Quantum simulation method, control layout for a quantum computer, method of determining same, and apparatus for quantum simulation

Abstract

A quantum simulation method is provided. The method includes receiving, as an input, a specification (902) of a quantum network protocol associated with a quantum network (100). The quantum network is a distributed network including a plurality of nodes (10a-f). A local quantum system (12a-f) is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local quantum evolution operations (20a, 20c, 50b), wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination thereof, is provided. Therein, according to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations (40ce), wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. Further, according to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations (32ab, 52df), wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. Further, according to (c), a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state (14cde, 34de) of K component quantum systems, wherein K is at least two. The method includes, in response to the receiving, encoding local quantum systems of the quantum network into at least a subset of quantum constituents (70) of a quantum computer (700), wherein each of the local quantum systems is disposed at a respective node of the quantum network during at least a portion of the quantum network protocol, wherein each of the local quantum systems is encoded into at least one quantum constituent. The method includes performing a simulation (950) of the quantum network protocol using the quantum computer, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.

Description

34296P-EP QUANTUM SIMULATION METHOD, CONTROL LAYOUT FOR A QUANTUM COMPUTER, METHOD OF DETERMINING SAME, AND APPARATUS FOR QUANTUM SIMULATION FIELD

[0001] Embodiments described herein relate to quantum networks. A quantum network is a distributed network including a plurality of nodes. At each node, a local quantum system may be provided. A quantum network protocol may be performed with respect to the quantum network. A quantum network protocol may include local quantum operations that may be performed at the respective nodes. Further, in a quantum network protocol, the nodes may communicate with each other by transmitting classical and / or quantum information. BACKGROUND

[0002] Quantum networks are quantum analogues of classical communication networks. A quantum network is a distributed network involving a plurality of nodes that are spatially separated from each other. Each node includes a local quantum system and an apparatus, e.g. a small-scale quantum computer, that may interact with the local quantum system to evolve the local quantum system locally. For example, local unitary operations and local measurements acting on the local quantum system, and more generally any local quantum evolution operation, may be performed. Due to the distance between the nodes, quantum interactions between local quantum systems disposed at different nodes, i.e. non-local quantum operations, may not be possible. On the other hand, the nodes may communicate with each other over arbitrary long distances, either by transmitting classical data (classical communication) or by transmitting quantum systems (quantum communication). A quantum network protocol may include arbitrary sequences of local quantum evolution operations, classical communication operations and quantum communication operations.

[0003] Quantum networks are thus enhanced versions of classical communication networks. The presence of quantum mechanical effects in quantum networks can be leveraged to improve the performance of classical networks. For example, entangled quantum states may be present between at least some of the nodes in a quantum network. Quantum entanglement is a resource that is purely quantum mechanical, i.e. has no classical counterpart. The presence of entanglement can allow a quantum network to perform complex tasks, which a classical34296P-EP communication network cannot achieve, or can only achieve at a very high computational overhead.

[0004] Understanding the full potential of quantum networks is a difficult task. Particularly, the possibilities as regards the number of nodes, the types of quantum systems that are considered both for the local quantum systems and the quantum information carriers that are used to transmit quantum information (i.e. whether they are qubits, d-level systems or infinite- dimensional systems), as well as the possibilities in designing different sequences of operations to form a quantum network protocol, are immense. Further, in real-life situations, quantum networks will be subject to noise, and it needs to be understood to which extent such noise will affect the performance of the quantum network protocols.

[0005] In light of the above, there is a need for improved methods to study the properties of quantum networks and quantum network protocols. SUMMARY

[0006] According to an embodiment, a quantum simulation method is provided. The method includes receiving, as an input, a specification of a quantum network protocol associated with a quantum network. The quantum network is a distributed network including a plurality of nodes. A local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local quantum evolution operations, wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination thereof, is provided, wherein (a), (b) and (c) are as follows. According to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations, wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. Further, according to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. Further, according to (c), a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a34296P-EP second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum systems, wherein K is at least two. The method includes, in response to the receiving, encoding local quantum systems of the quantum network into at least a subset of quantum constituents of a quantum computer, wherein each of the local quantum systems is disposed at a respective node of the quantum network during at least a portion of the quantum network protocol, wherein each of the local quantum systems is encoded into at least one quantum constituent. The method includes performing a simulation of the quantum network protocol using the quantum computer, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.

[0007] According to a further embodiment, an apparatus for performing a quantum simulation is provided. The apparatus includes a quantum computer. The quantum computer includes quantum constituents and a quantum evolution unit configured to perform an evolution of at least some of the quantum constituents. The apparatus includes a classical computing system. The classical computing system is configured for receiving, as an input, a specification of a quantum network protocol associated with a quantum network, wherein the quantum network is a distributed network comprising a plurality of nodes, wherein a local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local quantum evolution operations, wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination thereof, is provided, wherein (a), (b) and (c) are as follows. According to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations, wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. According to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. According to (c), a first local quantum system disposed at a first node of the quantum network includes a first34296P-EP component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum systems, wherein K is at least two. The classical computing system is configured for, in response to the receiving, encoding local quantum systems of the quantum network into at least a subset of the quantum constituents of the quantum computer, wherein each of the local quantum systems is disposed at a respective node of the quantum network during at least a portion of the quantum network protocol, wherein each of the local quantum systems is encoded into at least one quantum constituent. The quantum computer is configured for performing a simulation of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some of the quantum constituents from a first quantum state to a second quantum state using the quantum evolution unit to simulate the sequence of operations of the quantum network protocol.

[0008] According to a further embodiment, a method of determining a control layout for a quantum computer is provided. The method includes receiving, as an input, a specification of a quantum network protocol associated with a quantum network. The quantum network is a distributed network including a plurality of nodes, wherein a local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local quantum evolution operations, wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination thereof, is provided, wherein (a), (b) and (c) are as follows. According to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations, wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. Further, according to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. Further, according to (c), a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein34296P-EP the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum systems, wherein K is at least two. The method includes, in response to the receiving, determining a control layout for the quantum computer for performing a simulation of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.

[0009] According to a further embodiment, a control layout for a quantum computer for performing a simulation of a quantum network protocol associated with a quantum network is provided. The quantum network is a distributed network including a plurality of nodes. A local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local quantum evolution operations, wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination thereof, is provided, wherein (a), (b) and (c) are as follows. According to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations, wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. Further, according to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. Further, according to (c), a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum systems, wherein K is at least two. The control layout includes control instructions for the quantum computer, or information that allows determining said control instructions. The control instructions cause the quantum computer to perform a simulation of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some34296P-EP quantum constituents of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.

[0010] According to a further embodiment, a data carrier or data carrier signal carrying information representing the control layout according to embodiments described herein is provided.

[0011] Embodiments are also directed to methods for operating the systems described herein, and to the use of the systems to perform the methods according to the embodiments described herein.

[0012] Further advantages, features, aspects and details that can be combined with embodiments described herein are evident from the dependent claims, the description and the drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] A full and enabling disclosure to one of ordinary skill in the art is set forth more particularly in the remainder of the specification including reference to the accompanying drawings wherein: FIGS.1-6 show an exemplary quantum network protocol associated with a quantum network; FIG.7 shows a quantum computer including quantum constituents; FIG.8 show a quantum computer including subsystems, where each subsystem includes a group of quantum constituents; FIG.9 shows a quantum simulation of the quantum network protocol shown in Figs.1-6; FIG.10-11 show examples of a first portion of a quantum network protocol as described herein; FIG.12 shows a noisy sequence of quantum operations simulating a noisy portion of a quantum network protocol;34296P-EP FIG.13 shows an apparatus for performing a quantum simulation; FIG.14 illustrates a quantum simulation of a quantum network; FIG.15 illustrates different noise models (gate noise model and block noise model); FIGs.16(a)-(b) illustrate different ways for realizing a quantum channel on a quantum computer; FIGs.17(a)-(c) illustrate different ways for determining a noisy quantum circuit that realizes a desired quantum channel; FIGs.18(a)-(b) illustrate the performance of the building-block channel method described herein; FIGs.19(a)-(c) illustrate quantum circuits for realizing an amplitude damping channel in a noiseless (Fig.19(a)) or noisy setting (Figs. 19(b)- (c)); FIGs.20(a)-(c) illustrate the performance of the tailored circuit method described herein; and FIGs.21(a)-(c) illustrate the quantum simulation of a bit flip channel. DETAILED DESCRIPTION

[0014] Reference will now be made in detail to the various exemplary embodiments, one or more examples of which are illustrated in each figure. Each example is provided by way of explanation and is not meant as a limitation. For example, features illustrated or described as part of one embodiment can be used on or in conjunction with other embodiments to yield yet further embodiments. It is intended that the present disclosure includes such modifications and variations.

[0015] Within the description of the drawings, the same reference numbers refer to the same or similar components. Generally, only the differences with respect to the individual embodiments34296P-EP are described. The structures shown in the drawings are not necessarily depicted true to scale, and may contain details drawn in an exaggerated way to allow for a better understanding of the embodiments.

[0016] A quantum computer as described herein operates on quantum constituents. The quantum constituents are physical entities exhibiting quantum effects. That means, the quantum constituents are real-world objects. The quantum constituents may be called physical quantum constituents to highlight this property. Embodiments described here are not limited to any particular type of quantum constituents. For example, the quantum constituents may be atoms, ions, superconducting qubits, quantum dots, NV centers, photons, electrons, electromagnetic or mechanical resonators, and the like.

[0017] The quantum constituents can be regarded as smaller quantum systems that jointly form a larger physical quantum system. The physical quantum system can be in different quantum states, such as an initial quantum state (in which the physical quantum may be prepared at the beginning of a quantum computation) and a final quantum state (in which the physical quantum system may end up due to the quantum computation). The physical quantum system can be evolved from the initial quantum state to the final quantum state by performing a quantum evolution, e.g. by performing sequences of unitary operators and / or measurements. Such a quantum evolution is a real-world process, and particularly a controlled technical process (quantum computation) which brings the physical quantum system from the initial quantum state to an a priori unknown final quantum state that, in the context of the present disclosure, contains information about a quantum network protocol that is simulated by the quantum computer. Information regarding the simulated quantum network protocol can be revealed by measuring the physical quantum system or a part thereof, i.e., one or more of the quantum constituents. The act of measuring is also a physical / technical process. Measurements allow to obtain a read-out of the physical quantum system. A read-out of a physical quantum system is a set of one or more measurement values obtained by measurements of quantum constituents, involving physical interactions with the quantum constituents.

[0018] A quantum computer may include N quantum constituents, wherein N may be at least two, particularly N may be 5, 10, 20, 50, 100, 500, 1000, 10.000 or more. It shall be understood that the quantum systems shown in the figures and described in examples may be much smaller for illustrative and explanatory purposes, but shall not be understood to provide any limitation.34296P-EP

[0019] The quantum constituents may all be disposed in a same location, or the quantum computer may be a distributed quantum computer including subunits that are spatially separated from each other.

[0020] Embodiments described herein relate to quantum simulations of quantum network protocols. Before providing a detailed development of embodiments below, the concepts considered in the present disclosure are illustrated based on the examples shown in Figs.1-9. The figures in question show a relatively simple quantum network protocol and illustrate a possible quantum simulation thereof. These examples are merely provided for the sake of illustration and with the aim of helping the reader to understand the subject matter described herein, and the disclosure shall not be limited thereto.

[0021] Fig.1 shows a quantum network 100 including nodes 10a, 10b, 10c, 10d, 10e and 10f. The nodes are considered to be disposed at spatially distant locations. The number of nodes in a quantum network is considered to be arbitrary, and not limited to the number of nodes shown in Fig.1.

[0022] Each node may have a local quantum system disposed at the node for at least a certain period of time during a quantum network protocol performed by the quantum network 100. A local quantum system can be a quantum system of any type (e.g. may include qubits, d-level systems or infinite-dimensional systems, or any combination thereof). A local quantum system may itself include a group of smaller quantum systems, where the groups in question may contain an arbitrary number of such smaller quantum systems.

[0023] For example, as shown in Fig.1, the local quantum system at node 10a may be a single quantum system 12a, e.g. a qubit. The local quantum system at node 10b may be composed of two smaller quantum systems 12b (“component quantum systems”), each of which may, e.g., be a qubit. The local quantum system at node 10c may also be composed of two component quantum systems 12c. The local quantum system at node 10d may be composed of three component quantum systems 12d, e.g. three qubits. The local quantum system at node 10e may be a single quantum system 12e, which may, e.g., be an infinite-dimensional system. The local quantum system at node 10f may be composed of two component quantum systems 12f, e.g. two d-level systems with d > 2. These are merely examples for the purpose of illustration, and the disclosure shall not be limited thereto.34296P-EP

[0024] In the quantum network 100, it is considered that each node may include an apparatus, e.g. a small-scale quantum computer, to perform quantum operations (unitary operations, measurements, adiabatic evolutions, dissipative evolutions, and the like) on the local quantum system disposed at the node in question. Due to the distance between the nodes, it may be the case that entangling operations between multiple nodes are considered not feasible in a quantum network protocol. Further, a node may include a local classical communication system for transmitting classical data from the node to one or more other nodes of the quantum network and / or for receiving classical data one or more other nodes of the quantum network. Likewise, a node may include a local quantum communication system for transmitting quantum information carriers from the node to one or more other nodes and / or for receiving quantum information carriers from one or more other nodes.

[0025] Some of the local quantum systems of the quantum network may be in an entangled quantum state. This may already be the case at an initial stage, before the quantum network protocol has started. Alternatively, or additionally, it may be possible to generate an entangled quantum state as a part of the quantum network protocol, e.g. by creating the entangled quantum state locally at one particular node and by then distributing the entangled state over several nodes by transmitting one or more subsystems of the entangled state to one or more respective nodes. Fig.1 exemplarily shows an entangled quantum state 14cde (represented schematically by a triangle) between nodes 10c, 10d and 10e. The entangled quantum state 14cde is a state of three quantum systems, namely a first quantum system 12c at node 10c, a second quantum system 12d at node 10d, and a third quantum system 12e at node 10e. Again, this is merely an illustrative example. The subset of nodes that is involved in the entangled quantum state can be an arbitrary subset. Further, apart from an entangled quantum state, another type of quantum state may be provided, e.g. a separable mixed state.

[0026] Figs.2-6 illustrate an exemplary quantum network protocol associated with the quantum network 100. The quantum network protocol includes a sequence of operations.

[0027] In a first operation of the quantum network protocol, shown in Fig.2, a local quantum evolution operation 20a on the quantum system 12a is performed at node 10a. The local quantum evolution operation 20a may act only on the quantum system 12a. A local quantum evolution operation may be any possible quantum evolution (unitary operation, measurement, adiabatic evolution, and the like) acting on the local quantum system in question.34296P-EP

[0028] In a second operation of the quantum network protocol, also shown in Fig. 2, a local quantum evolution operation 20c is performed on a portion of the local quantum system at node 10c, namely on one of the component quantum systems 12c. Generally, the local quantum evolution operation 20c may again be any possible quantum evolution. In the present example, for the sake of illustration, it is considered that the local quantum evolution operation 20c may be a measurement of the component quantum system 12c in question. The measurement may disentangle the component quantum system 12c from the quantum systems 12d and 12e with which the quantum system 12c was initially entangled within the entangled quantum state 14cde. As a result, the entangled quantum state 14cde may be transformed into another entangled quantum state 34de involving the local quantum systems at nodes 10d and 10e, as shown in Fig. 3, where the entangled quantum state 34de is however not entangled with the local quantum system at node 10c.

[0029] In a third operation of the quantum network protocol, as shown in Fig. 3, a quantum communication operation 32ab may be performed, wherein the quantum system 12a may be transmitted from node 10a to node 10b over a quantum communication channel. In other words, the quantum system 12 is physically sent from node 10a to node 10b. Accordingly, thereafter, the quantum system 12a is part of the local quantum system at node 10b, in addition to the two quantum systems 12b, as shown in Fig.4. Further, the node 10a no longer includes the quantum system 12a, and in fact no longer includes a local quantum system at all.

[0030] In a fourth operation of the quantum network protocol, as shown in Fig. 4, a classical communication operation 40ce may be performed, wherein classical data is transmitted from node 10c to node 10e. For example, the outcome of the measurement that was performed at node 10c in the second operation of the quantum network protocol may be sent from node 10c to node 10e.

[0031] In a fifth operation of the quantum network protocol, shown in Fig.5, a local quantum evolution operation 50b may be performed at node 10b, wherein the local quantum evolution operation 50b acts jointly on the three quantum systems 12a and 12b that are disposed at the node 10b.

[0032] In a sixth operation of the quantum network protocol, as also shown in Fig.5, a quantum communication operation 52df may be performed, wherein a quantum system 12d is transmitted from node 10d to node 10f over a quantum communication channel. Accordingly, thereafter,34296P-EP the quantum system 12d in question is part of the local quantum system at node 10f, in addition to the two quantum systems 12f, as shown in Fig.6. Accordingly, the entangled quantum state 34de is now provided between nodes 10e and 10f.

[0033] According to embodiments described herein, a quantum simulation of a quantum network protocol is performed by a quantum computer. A quantum simulation is performed on a quantum computer and is distinguished from a classical simulation, since the latter involves classical information processing only.

[0034] Fig. 7 shows a schematic representation of a quantum computer 700. The quantum computer includes quantum constituents 70. A quantum constituent 70 can be any kind of quantum system. For example, the quantum constituents 70 can be qubits, yet this is merely an example and the disclosure is not limited thereto. There is no particular limitation on the number of quantum constituents 70 that are included in the quantum computer 700. The quantum constituents 70 may all be of a same type (e.g. all of them being qubits), or quantum constituents of different types may be part of the quantum computer 700.

[0035] A quantum network protocol associated with a quantum network may be simulated using the quantum computer 700. It may be the case that the quantum network itself is not physically provided. For example, with respect to Figs.1-6, it may be the case that the quantum systems 12a-f of the quantum network 100 are not physically realized. In contrast, the quantum constituents 70 of the quantum computer 700 are real-life physical systems that are engineered and controlled to simulate the quantum network protocol. By way of the simulation, the properties of the quantum network protocol can be studied, without a need to physically realize the actual quantum network protocol. Particularly, the quantum computer 700 may be a programmable quantum computer that is configured to simulate a plurality of different quantum network protocols, such as respective quantum network protocols where local quantum systems of different types are used (e.g. qubits versus infinite-dimensional systems), respective quantum network protocols where the involved quantum systems (local quantum systems, quantum information carriers) are subject to different types / strengths of noise, and the like. Accordingly, the same apparatus, i.e. the quantum computer 700, can be used to study the properties of such different quantum network protocols, without the need to physically realize each individual quantum network.34296P-EP

[0036] In order to simulate a quantum network on the quantum computer 700, the quantum network 100 may be encoded into the quantum constituents 70. In the example under consideration, the quantum systems 12a-f, which form the quantum systems that are present in the quantum network at the start of the quantum network protocol (i.e. the configuration shown in Fig. 1), may be encoded into the quantum constituents 70. The encoding may be updated along the course of the quantum simulation, for example if new quantum systems are added, if quantum systems are removed, or if quantum systems are transmitted from one node to another.

[0037] There are several possible ways to encode the quantum systems 12a-f into the quantum constituents 70, as discussed in more detail below. One possible example involves a “local” encoding, wherein each of the quantum systems 12a-f is encoded into a corresponding subsystem of the quantum computer 700, where each subsystem is a single quantum constituent 70 or a group of several quantum constituents 70. For example, as shown in Fig.8, the quantum computer 700 may include disjoint subsystems 80a-f. Each of the quantum systems 12a-f of the quantum network 100 may be encoded into a respective one of the subsystems 80a-f. For example, quantum system 12a at node 10a may be encoded into a group of two quantum constituents 70 forming a subsystem 80a. Each of the two quantum systems 12b at node 10b may be encoded into a group of two quantum constituents 70 included in a subsystem 80b of the quantum computer 700, so that the subsystem 80b includes four quantum constituents 70 in total; and so on. The number of quantum constituents in the subsystems 80a-f shown in Fig.8 are merely examples and the disclosure shall not be limited thereto. As shown in Fig. 8, the quantum computer may include additional quantum constituents (ancillary constituents), which are not part of a subsystem encoding a portion of the quantum network.

[0038] In order to encode a quantum system of the quantum network 100 into a subsystem of the quantum computer 700, the quantum system may be represented as a d-dimensional Hilbert space (where d may be finite or infinite), and an identification can be made between the quantum states in said Hilbert space and corresponding quantum states in the subsystem under consideration (such an identification is possible as long as the dimension of the subsystem is at least d), thus providing the desired encoding. Again, this is merely one possible example of an encoding, and the disclosure is not limited thereto. Embodiments described herein do not depend on a particular type of encoding being used.34296P-EP

[0039] As illustrated in Fig.9, a quantum simulation 950 of the quantum network protocol may be performed by the quantum computer 700.

[0040] A specification 902 of the quantum network protocol may be received, as an input, by a classical computing system 900 that is connected to the quantum computer 700. The specification 902 may include information describing the components of the quantum network 100 (e.g. the nodes 10a-f, the quantum systems 12a-f, and any quantum states that are initially provided, such as the quantum state 14cde) and the operations performed during the quantum network protocol (e.g. the first to sixth operations illustrated in Figs.2-6).

[0041] Based on the specification 902, an encoding of the quantum systems 12a-f into the quantum constituents 70, such as the encoding described with respect to Fig. 8, may be determined, e.g. by the classical computing system 900.

[0042] At the start of the quantum simulation, each quantum constituent 70 may be prepared in a default initial quantum state (e.g. the state |0>). A quantum operation 910cde (e.g. a unitary operation) acting on subsystems 80c, 80d and 80e may be performed by the quantum computer 700 in order to prepare these subsystems in a quantum state corresponding, via the encoding, to the entangled quantum state 14cde, thereby preparing the quantum computer 700 in an initial configuration corresponding to the initial configuration of the quantum network 100 as shown in Fig.1.

[0043] The operations performed during the quantum network protocol may be mapped, based on the encoding, to corresponding operations that can be performed by the quantum computer 700 (or in some cases by the classical computing system 900). For example, the first to sixth operations of the quantum network protocol, as described above, may be mapped to corresponding operations 920a, 920c, 930ab, 940ce, 950b and 950df, respectively. These mappings, which are induced by the encoding under consideration, may be determined, for example, by the classical computing system 900. Operation 920a, acting on subsystem 80a, simulates local quantum evolution operation 20a (first operation of the quantum network protocol). Operation 920c, acting on subsystem 80c, simulates local quantum operation 20c (second operation of the quantum network protocol). Operation 930ab, acting on subsystems 80a and 80b, simulates quantum communication operation 32ab (third operation of the quantum network protocol). Operation 940ce, performed by the classical computing system 900, simulates classical communication operation 40ce (fourth operation of the quantum network34296P-EP protocol); and so on. Accordingly, a quantum simulation of the quantum network protocol is provided.

[0044] In the example discussed above, each operation of the quantum network protocol is simulated by a respective operation performed by the quantum computer. The disclosure is not limited thereto. An operation of the quantum network protocol may also be simulated by a sequence of several operations performed by the quantum computer. Further, several operations performed in the quantum network protocol may be grouped, and the quantum computer may be performed a quantum simulation of such a group of operations as a whole (“event-based simulation”).

[0045] For the sake of simplicity, the operations 910cde to 950df shown in Fig.9 act only on the respective subsystems 80a-f. The disclosure is not limited thereto. Particularly, the operations in question may also act on additional quantum constituents 70 of the quantum computer 700, i.e. ancillary constituents that do not necessarily correspond to an associated quantum system of the quantum network 100.

[0046] After the quantum operations 910cde to 950df have been performed, one or more measurements 960 may be performed by the quantum computer 700 to obtain a read-out of the quantum simulation. A measurement may include a measurement of one or more quantum constituents 70. The read-out may reveal information regarding the quantum network protocol that is being studied, and may hence be helpful to determine a priori unknown properties of the quantum network protocol. In the example shown in Fig.9, (one or more quantum constituents 70 belonging to) the subsystems 80e and 80e are measured. Again, this is merely an illustrative example. It shall be understood that some quantum simulations might not include any measurement aimed to obtain a read-out. Instead, one or more quantum constituents 70 that are in a final quantum state of the quantum simulation may be transmitted over a quantum communication channel for further analysis by a different user.

[0047] According to an embodiment, a quantum simulation method is provided. The method includes receiving, as an input, a specification of a quantum network protocol associated with a quantum network. The quantum network is a distributed network including a plurality of nodes. A local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local34296P-EP quantum evolution operations, wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination thereof, is provided. Therein, according to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations, wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. Further, according to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. Further, according to (c), a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum systems, wherein K is at least two. The method includes, in response to the receiving, encoding local quantum systems of the quantum network into at least a subset of quantum constituents of a quantum computer, wherein each of the local quantum systems is disposed at a respective node of the quantum network during at least a portion of the quantum network protocol, wherein each of the local quantum systems is encoded into at least one quantum constituent. The method includes performing a simulation of the quantum network protocol using the quantum computer, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.

[0048] Embodiments described herein provide the advantage that the properties of quantum networks and quantum network protocols can be studied based on the quantum simulation, thereby providing a convenient and flexible tool to design new quantum network protocols and investigate the potential thereof.

[0049] Embodiments described herein differ from conventional approaches to simulate quantum network protocols, which are based on classical simulations only. A quantum simulation, as considered herein, has the advantage that the exponentially large dimension of the total quantum system formed by the local quantum systems of the quantum network34296P-EP provides no hindrance to the quantum simulation, since the quantum computer is itself a large quantum system. Accordingly, arbitrarily large and complex quantum networks may be simulated by the quantum simulation method described herein. In contrast, classical simulation methods are inherently limited in their capability to simulate large quantum networks, in view of the exponentially large amount of information that needs to be stored and processed in order to represent the quantum state of a large quantum network classically.

[0050] It is further noted that quantum simulation methods are conventionally used in other areas, namely to simulate particular physical interactions arising in nature, e.g. strongly correlated systems, condensed matter systems, field theories or quantum chemistry. Quantum networks are different from these situations, since quantum networks involve man-made protocols that are designed to perform a particular distributed communication task. Further, the nodes of a quantum network, different from e.g. a lattice of spins, can themselves include complex quantum systems involving multiple qubits (or other quantum systems) and apparatuses to act thereon (e.g. small-scale quantum computers), As described above, the conventional approach to study the properties of quantum networks is to rely on classical simulation methods. However, the present inventors have realized that the use of quantum simulation methods can significantly improve the simulation of quantum network protocols as compared to classical simulation methods.

[0051] A specification of a quantum network protocol (e.g. specification 902 in Fig. 9) may include information describing the quantum network protocol or a portion thereof. A specification of a quantum network protocol may include information that is configured for allowing the simulation of the quantum network protocol to be performed by the quantum computer. A specification of a quantum network protocol may include information allowing an operator to determine which operations are included in the sequence of operations of the quantum network protocol. In this context, the term “specification” can be understood in the sense of “description” or “(classical) information”.

[0052] A specification of a quantum network protocol may be a specification of the entire quantum network protocol or of a portion thereof. A specification of a quantum network protocol may include any one of the following, or any combination thereof: a specification of the nodes of the quantum network, or a portion of the nodes; a specification of one or more local quantum systems that are included in the quantum network; a specification of one of more34296P-EP quantum states that may be included in the quantum network, such as one or more joint quantum states (e.g. entangled quantum states) distributed across nodes of the quantum network; a specification of one or more apparatuses disposed at respective nodes to interact with the local quantum systems; a specification of one or more systems to transmit / receive classical information and / or quantum information carriers; a specification of one or more local quantum evolution operations that may be included in the quantum network protocol; a specification of one or more classical communication operations that may be included in the quantum network protocol; a specification of one or more quantum communication operations that may be included in the quantum network protocol; and the like.

[0053] The quantum network protocol may be specified abstractly, i.e. without referring to a particular physical implementation of the local quantum systems of the quantum network. For example, if the local quantum systems include qubits, the specification of the quantum network protocol may include a specification of the qubits as abstract mathematical entities (namely, as two-dimensional Hilbert spaces), without referring to a particular physical implementation of the qubits. This may be of interest in situations where implementation-independent properties of the quantum network protocol shall be simulated by the quantum computer.

[0054] Alternatively, an aim of the quantum simulation may be to simulate the quantum network protocol for a particular physical implementation thereof. For example, the quantum simulation may simulate a realization of the quantum network by means of superconducting qubits. In such a case, the specification of the quantum network protocol may include information regarding the chosen physical implementation of the local quantum systems (e.g. whether the local systems are realized by photons, ions, superconductors, or the like) and / or of the quantum operations performed during the quantum network protocol. In any case, embodiments described herein are not restricted to any particular physical implementation and allow for a quantum simulation of quantum networks having arbitrary physical implementations. Further, as described above, the quantum constituents of the quantum computer can also be arbitrary quantum systems, i.e. embodiments described herein do not rely on a particular physical implementation of the quantum constituents. Particularly, a physical implementation of the quantum constituents may be different from a physical implementation of the quantum network (e.g. a quantum computer having superconducting quantum constituents may be used to simulate a quantum network based on ions and photons).34296P-EP

[0055] Receiving the specification of the quantum network protocol may include any arbitrary manner of acquiring the specification, e.g. by reading the specification from a memory, by receiving the specification via a transmission, or the like. That the specification is received as an input can be understood in the sense that embodiments described herein are capable of receiving a specification of an arbitrary quantum network protocol and perform a simulation thereof. The received specification of a quantum network protocol is a variable input. The quantum network protocol that is to be simulated may be a priori unknown. Embodiments described herein thereby differ from systems that are tailored (e.g. hard-wired) for specifically simulating one fixed quantum network protocol. In the latter systems, the quantum network protocol is not received as a variable input.

[0056] According to embodiments, the quantum network is a distributed network comprising a plurality of nodes. The plurality of nodes may include 2, 3, 4, 5, 10, 100 or more nodes. The nodes may be spaced apart from each other. There is no particular limitation on the distances between the nodes. Yet it is considered that the nodes are spaced apart from each other in a manner that may not allow for quantum interactions between local quantum systems disposed at different nodes of the quantum network. It may be the case that quantum interactions between local quantum systems disposed at different nodes of the quantum network (also called non- local quantum interactions) are not part of a quantum network protocol.

[0057] A node of a quantum network can be understood as a station, location, and the like, where quantum operations may be performed. A node may include a local quantum system, which may itself be a composition of several individual quantum systems (called component quantum systems herein). A node may include a quantum device (or multiple quantum devices) to act on the local quantum system. A node may include a transmission system for transmitting classical data and / or quantum information carriers. A node may include a reception system for receiving classical data and / or quantum information carriers.

[0058] A local quantum system disposed at a node of the quantum network may be a quantum system that is spatially localized at the node in question. That the local quantum system is “local” can include that the local quantum system does not include a portion disposed at another node of the quantum network.

[0059] A local quantum system disposed at a node of the quantum network may include an arbitrary quantum system, e.g. a qubit, a d-level system with arbitrary d, or an infinite-34296P-EP dimensional quantum system. A local quantum system may include a group of several component quantum systems (as is e.g. the case for nodes 10b, 10c, 10d and 10f in Fig.1), each of which may be an individual quantum system, possibly of different types, e.g. a group including multiple qubits, a group including one or more qubits and one or more infinite- dimensional system, and the like. One or more component quantum systems disposed at a node may be auxiliary quantum systems. These are merely examples and the disclosure is not limited thereto. The type and number of component quantum systems included in a local quantum system is arbitrary. Further, as described above, there is no limitation as regards the physical implementation of the quantum systems under consideration.

[0060] According to embodiments, a local quantum system is disposed at each respective node during at least a portion of the quantum network protocol. The properties of a local quantum system (e.g. the dimension, the number of component quantum systems and / or the type of component quantum systems of the local quantum system) that is disposed at a given node during the quantum network protocol, and whether a local quantum system is even disposed at the node at all, may change during the quantum network protocol. One or more component quantum systems may be added and / or removed at a node along the course of the quantum network protocol. In one example, it may be the case that a local quantum system at a node is measured at a certain time during the quantum network protocol, and that after the measurement the local quantum system is not needed anymore for the remainder of the quantum network protocol, so that the local quantum system may be discarded after the measurement. In another example, at a certain time during the quantum network protocol a local quantum system (or a portion thereof) may be sent from a first node to a second node, causing the local quantum system (or the portion in question) to be removed from the first node and to be introduced at the second node, thus leading to a change in the local quantum system at both nodes (see e.g. quantum communication operation 32ab in Fig.3). It shall be understood that these are merely a few examples of situations where the properties of a local quantum system may change during a quantum network protocol, and the disclosure is not limited thereto.

[0061] According to embodiments, a sequence of operations of a quantum network protocol includes one or more local quantum evolution operations.

[0062] A local quantum evolution operation (see e.g. local quantum evolution operations 20a, 20c and 50b) evolves a local quantum system disposed at one of the nodes of the quantum34296P-EP network. The type of quantum evolution may be arbitrary. For example, a local quantum evolution operation may include a unitary operation, a measurement, an adiabatic evolution, a dissipative evolution or any other kind of quantum evolution, and any combination thereof. A local quantum evolution operation that is a part of a quantum network protocol to be simulated may be an ideal evolution which is free of noise or an evolution subject to noise.

[0063] It may be the case that a local quantum evolution operation acts on a single local quantum system at a particular node of the quantum network. It may be the case that a local quantum evolution operation does not entangle multiple (i.e. two or more) local quantum systems disposed at different nodes of the quantum network with each other. It may be the case that, where a local quantum operation acts on a first local quantum system disposed at a first node of the quantum network, the local quantum operation does not additionally act on any other node of the quantum network. Still, it is permitted that the local quantum evolution operation acts on multiple quantum systems disposed at the first node (as e.g. local quantum evolution 50b in Fig.5).

[0064] The one or more local quantum evolution operations that are included in a sequence of operations of a quantum network protocol may include a single local quantum evolution operation, a plurality of local quantum evolution operations acting at a same node or a plurality of local quantum evolution operations acting at different nodes. The one or more local quantum evolution operations may include 1, 2, 3, 4, 5, 10, 100 or more local quantum evolution operations.

[0065] A sequence of operations of a quantum network protocol may include one or more classical communication operations. A classical communication operation (see e.g. classical communication operation 40ce in Fig. 4) is an operation where classical information is transmitted and / or received. Classical information, which may also be called classical data, may be understood as information that is carried by classical information carriers, such as classical bits or the like. A classical communication operation may involve a transmission of classical information only. It may be the case that a classical communication operation does not include a transmission of quantum information carriers. A classical communication operation may include transmitting classical information from a first node of the quantum network to a second node of the quantum network. The classical information may be transmitted from the first node to the second node and optionally to additional nodes of the quantum network, so that the same34296P-EP information is transmitted to several nodes. The classical information may be transmitted from the first node using a transmitter. The classical information may be received at the second node, and at any further nodes, using a receiver.

[0066] The one or more classical communication operations that are included in a sequence of operations of a quantum network protocol may include a single classical communication operation, a plurality of classical communication operations involving a transmission of classical information from a same first node to a same second node or a same set of second nodes, or a plurality of classical communication operations involving a transmission of classical information between different respective nodes or sets of nodes. The one or more classical communication operations may include 1, 2, 3, 4, 5, 10, 100 or more classical communication operations.

[0067] A sequence of operations of a quantum network protocol may include one or more quantum communication operations. A quantum communication operation (see e.g. quantum communication operations 32ab and 52df) is an operation where one or more quantum information carriers are transmitted and / or received. A quantum information carrier may be a quantum system (e.g. a photon, ion, or any other quantum system) that is configured for encoding information into a quantum state |^> of the quantum system. In a quantum communication operation, the quantum system may be physically transmitted from a first node of the quantum network to a second node of the quantum network, thereby transmitting the quantum state and thus the information contained therein, from the first node to the second node. For example, a quantum communication operation may include sending one or more photons from a first node to a second node. A quantum information carrier may be configured for encoding information into a superposition of quantum states, e.g. superpositions of the form = a |0> + b |1>, where a and b are complex coefficients and |0> and |1> are quantum basis states (computational basis states) of a qubit; the latter is merely a simple example for explanatory purposes, and arbitrary superpositions of arbitrary quantum systems may be considered.

[0068] A quantum information carrier is distinguished from a quantum system that is merely used for transmitting classical information. For example, classical bits may be transmitted by means of electromagnetic waves using conventional (i.e. classical) data transmission systems. Electromagnetic waves are ultimately quantum mechanical systems, yet the information being34296P-EP transmitted is purely classical, in particular such systems are not capable of transmitting information encoded into superpositions of quantum states. Accordingly, such conventional data transmission systems do not involve a transmission of quantum information carriers and do not perform quantum communication operations as described herein.

[0069] A quantum communication operation may include transmitting one or more quantum information carriers from a first node of the quantum network to a second node of the quantum network. Before said transmission, the one or more quantum information carriers may be part of a first local quantum system disposed at the first node. After the transmission, the one or more quantum information carriers may be part of a second local quantum system disposed at the second node. The one or more quantum information carriers may be transmitted from the first node to the second node and optionally to additional nodes of the quantum network, so that the same information is transmitted to several nodes. The one or more quantum information carriers may be transmitted from the first node using a quantum transmission system, or quantum transmitter. The one or more quantum information carriers may be received at the second node, and at any further nodes, using a quantum receiving system, or quantum receiver. Examples of quantum transmitters and quantum receivers are atoms or ions stored in a cavity which transmit or receive photons, quantum dots, nitrogen-vacancy centers (NV centers) or superconducting circuits.

[0070] The one or more quantum communication operations that are included in a sequence of operations of a quantum network protocol may include a single quantum communication operation, a plurality of quantum communication operations involving a transmission of quantum information carriers from a same first node to a same second node or a same set of second nodes, or a plurality of quantum communication operations involving a transmission of quantum information carriers between different respective nodes or sets of nodes of the quantum network. The one or more quantum communication operations may include 1, 2, 3, 4, 5, 10, 100 or more quantum communication operations. A quantum communication operation that is a part of a quantum network protocol to be simulated may be an ideal quantum communication operation which is free of noise or a quantum communication operation subject to noise.

[0071] A first local quantum system disposed at a first node of the quantum network may include a first component quantum system. A second local quantum system disposed at a second34296P-EP node of the quantum network may include a second component quantum system. The first component quantum system and the second component quantum system may form part of a joint quantum state of K component quantum systems, wherein K is at least two.

[0072] The first component quantum system can be an arbitrary quantum system included in the first local quantum system. The first component quantum system may be the entire first local quantum system or a portion thereof. In one example, the first local quantum system may include a group of individual quantum systems (e.g. a group of n qubits), wherein the first component quantum system may be one of the individual quantum systems (e.g. a single qubit), several of the individual quantum systems (e.g. a subgroup of m qubits, with m < n) or even all of the individual quantum systems comprised in the first local quantum system. In another example, the first local quantum system may consist of a single, non-composite quantum system, and the first component quantum system is equal to the first local quantum system. Analogous considerations apply to the second component quantum system and the second local quantum system.

[0073] The joint quantum state of the K component quantum systems may be an entangled quantum state of the K component quantum systems (see e.g. entangled quantum state 14cde in Fig.1). Alternatively, the joint quantum state may be a separable state, e.g. a separable mixed state, of the K component quantum systems. The number K of component quantum systems may be 2 or more, 3 or more, 4 or more, 5 or more, 10 or more, or 100 or more. At least some, or even all, of the K component quantum systems may be included in respective local quantum systems disposed at respective nodes of the quantum network.

[0074] During the quantum network protocol, several joint quantum states as described above may be provided. At different times during the quantum network protocol, the nodes that are involved in the joint quantum state, the number K of component quantum systems, and the particular kind of joint quantum state that is provided may vary.

[0075] The sequence of operations of the quantum network protocol may include a local quantum evolution operation performed on a first local quantum system disposed at a first node of the quantum network. The sequence of operations may include a classical communication operation that transmits classical information from the first node to a second node of the quantum network. The classical communication operation may be performed in response to the local quantum evolution operation performed on the first local quantum system. For example,34296P-EP the local quantum evolution operation may include a measurement of the first local quantum system, and the classical communication operation may transmit classical information regarding an outcome of the measurement from the first node to the second node.

[0076] Additionally or alternatively, the sequence of operations of the quantum network protocol may include a local quantum evolution operation performed on a first local quantum system disposed at a first node of the quantum network. The sequence of operations may include a quantum communication operation that transmits one or more quantum information carriers from the first node to a second node of the quantum network. The quantum communication operation may be performed in response to the local quantum evolution operation performed on the first local quantum system. For example, the local quantum evolution operation may include a unitary evolution of the first local quantum system, and, in response thereto, the quantum communication operation may transmit the first local quantum system or a portion thereof from the first node to the second node.

[0077] Additionally or alternatively, a first local quantum system disposed at a first node of the quantum network may include a first component quantum system and a second local quantum system disposed at a second node of the quantum network may include a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum systems, wherein K is at least two. The sequence of operations of the quantum network protocol may include a local quantum evolution operation performed on at least one of the first component quantum system and the second component quantum system. The local quantum evolution operation may be performed in response to providing the joint quantum state. For example, a first qubit and a second qubit may be disposed at the first node and the second node, respectively. The first qubit and the second qubit may be part of an entangled state of K qubits, wherein each of the K qubits is disposed at one of the nodes of the quantum network, and wherein K is two or larger. A local quantum evolution operation, which may, for example, be a unitary operation, may be performed on, say, the first qubit while the entangled state of the K qubits is provided.

[0078] The sequence of operations of a quantum network protocol may include one or more local quantum evolution operations, one or more classical communication operations and one or more quantum communication operations in any number and combination. Further, such operations may be combined arbitrarily with the provision of joint quantum states of component34296P-EP quantum systems. It is also possible that at least one of classical communication operations, quantum communication operations, or the provision of joint quantum states is absent from a quantum network protocol.

[0079] In response to receiving the specification of the quantum network protocol as an input, local quantum systems of the quantum network are encoded into quantum constituents of the quantum computer. Each of the local quantum systems is disposed at a respective node of the quantum network during at least a portion of the quantum network protocol. Each of the local quantum systems is encoded into at least one quantum constituent.

[0080] An encoding of a local quantum system of the quantum network into one or more quantum constituents of the quantum computer can be understood as a mapping which maps the local quantum system, or at least certain properties of the local quantum system, to the one or more quantum constituents. Due to the encoding, the one or more quantum constituents may behave like the local quantum system, at least as regards properties of the local quantum system that are relevant for the quantum network protocol. Information contained in the local quantum system may be mapped, via the encoding, onto the one or more quantum constituents. An operation performed on the local quantum system (e.g. a local quantum evolution operation) during the quantum network protocol may be simulated, via the encoding, by performing a corresponding operation (or sequence of operations) on the one or more quantum constituents.

[0081] A local quantum system of the quantum network may be encoded into a single quantum constituent or in a set of several quantum constituents of the quantum computer. The latter set may include some or all of the quantum constituents of the quantum computer.

[0082] A local quantum system of dimension d (where d is at least two, including the possibility that d is infinite) may be encoded into a subsystem of the quantum computer, wherein the subsystem may include one or more quantum constituents (see e.g. subsystems 80a-f in Fig.8). The local quantum system and the quantum constituent(s) of the subsystem may be quantum systems of a same type (e.g. both being qubit systems) or a different type (e.g. the local quantum system is a d-dimensional system with d > 2 while the subsystem consists of qubits). If the entire quantum state of the d-dimensional local quantum system is to be encoded into the subsystem in question, then the total dimension of the subsystem (i.e. the sum of the dimensions of any quantum constituents that are comprised in the subsystem) shall be at least d. For example, if a local quantum system is a qubit (thus d = 2), and if the quantum constituents of34296P-EP the quantum computer are also qubits, the local quantum system may be encoded into a single quantum constituent (i.e. a single qubit) in a trivial manner, or may be encoded in a subsystem comprising several quantum constituents (i.e. several qubits), e.g. by associating the local quantum system with a two-dimensional subspace of the subsystem. If, on the other hand, the dimension of the local quantum system is equal to three (i.e. the local quantum system is a qutrit) while the quantum constituents are qubits, the local quantum system may be encoded into a subsystem comprising at least two qubits. These are merely simple examples to illustrate the concepts involved.

[0083] For the purpose of the quantum simulation of the quantum network protocol, it may be sufficient to encode only particular properties of a local quantum system into a subsystem of the quantum computer (which may be called a “partial encoding”), rather than encoding the entire quantum state of the local quantum system (“complete encoding”) as described above. For example, if the local quantum system is infinite-dimensional, or d-dimensional with d very large but finite, it may be sufficient to encode only some of the quantum levels of the local quantum system into a subsystem of the quantum computer. (For example, in the example of an infinite-dimensional local quantum system, it may be the case that only the first m levels of the local quantum system are physically relevant, while the higher levels involve unrealistically high energies.) In such a case, the local quantum system can be encoded into a subsystem having a dimension which can be smaller than the dimension of the local quantum system, by encoding only the relevant quantum levels into the subsystem.

[0084] There are several possibilities for encoding a set of several local subsystems of the quantum network into the constituents of the quantum computer.

[0085] In one example, the quantum computer may include L disjoint subsystems of quantum constituents, for some L larger than or equal to two, wherein each subsystem includes at least one quantum constituent. For example, the quantum constituents within a same subsystem may be spatially close to each other, e.g. nearest neighbors, next-nearest neighbors or the like. The number of quantum constituents with a subsystem may be relatively small. Each subsystem may provide an encoding of a corresponding local quantum system of the quantum network. That is to say, in this example, L local quantum systems may be encoded into L disjoint, localized groups of quantum constituents. The encoding of the present example may be called a local encoding. A local encoding is illustrated in Fig.8.34296P-EP

[0086] More specifically, in one example of a local encoding of a local quantum system of a quantum network into a subsystem of the quantum computer, quantum states |0>, |1>, ... may form an orthogonal quantum basis of the local quantum system. The associated subsystem of the quantum computer may have orthogonal quantum states |^0>, |^1>, ...., wherein each |^i> is a quantum state of the subsystem (for example, if the subsystem consists of k qubits, then each |^i> may be a k-qubit state). The local quantum system may be encoded into the subsystem by associating |^0> with |0>, by associating |^1> with |1>, and so on. Thus, according to the encoding under consideration, a quantum state= a |0> + b |1> + ... of the local quantum system of the quantum network (where a, b, ... are complex coefficients) is mapped onto a quantum state+ .... of the subsystem of the quantum computer. This is merely one possible example of a local encoding, and the disclosure shall not be limited thereto.

[0087] In another example, several local quantum systems of the quantum network may be encoded in a same group of quantum constituents. The latter group may be relatively large, and may even consist of substantially all quantum constituents of the quantum computer. In other words, the information contained in a local quantum system may be spread across a large number of quantum constituents, and the same group of quantum constituents may encode information regarding several, possibly even all, local subsystems of the quantum network. Such a situation may arise, for example, when starting out from the preceding example, i.e. a local encoding wherein the local quantum systems are encoded into L disjoint, localized subsystems of quantum constituents, supplemented by a “global” unitary operation acting on M quantum constituents jointly, wherein M may have an order of magnitude similar to the total number of quantum constituents of the quantum computer. Such a unitary operation (which may be called a scrambling unitary operation) scrambles the initial local encoding and distributes the information initially contained locally in each of the L subsystems across the M quantum constituents. By inverting the scrambling unitary operation at a later stage, e.g. at the end of the simulation, the initial local encoding may be restored. The encoding in the present example may be called a global encoding.

[0088] The above are merely two examples of possible encodings, and the disclosure is not limited thereto. Generally, the quantum simulation method and apparatus described herein do not rely on a particular encoding or set of encodings, i.e. arbitrary encodings are permitted.34296P-EP

[0089] According to embodiments, at a time T during the quantum network protocol, one or more local quantum systems may be disposed at one or more respective nodes of the quantum network. The one or more local quantum systems may be in a first quantum state (e.g. a quantum state or more generally a mixed state ^) at the time T. At a time T’ during the quantum simulation of the quantum network protocol, the one or more local quantum systems may be encoded into at least a portion of the quantum constituents of the quantum computer. At the time T’, at least a portion of the quantum constituents may be in a second quantum state (e.g. a quantum state or more generally a mixed state ^’) that encodes at least one property of the first quantum state. In some cases, the entire first quantum state may be encoded into the second quantum state (“complete encoding”). In other cases, only a subset of properties of the first quantum state may be encoded into the second quantum state (“partial encoding”).

[0090] Particularly, the encoding may be a local encoding as described herein. The one or more local quantum systems may include M local quantum systems, wherein M is two or larger. The quantum computer may include M subsystems each including one or more quantum constituents. The M subsystems may be mutually disjoint. In the second quantum state, each of the M local quantum systems of the quantum network may be encoded into a respective subsystem of the M subsystems.

[0091] For example, the quantum network shown in Fig.1 may be in a quantum state |^1> at a time T1 during the quantum network protocol. At a corresponding time T1’ during the quantum simulation, particularly a time directly after operation 910cde has been performed (see Fig. 9), the quantum constituents 70 may be in a quantum stateencoding at least one property of the quantum stateFurther, the quantum network shown in Fig.2 may be in a quantum state |^2> at a time T2 during the quantum network protocol. At a corresponding time T2’ during the quantum simulation, particularly a time directly after operations 920a and 920c have been performed (see Fig.9), the quantum constituents 70 may be in a quantum state |^2’> encoding at least one property of the quantum state |^2>; and so on.

[0092] As described above, the properties (dimension, type of quantum system, and the like) of a local quantum system disposed at a given node of the quantum network, and whether a local quantum system is disposed at a given node in the first place, may vary throughout the quantum network protocol. Correspondingly, also the encoding that is used to map the local quantum systems to the quantum constituents of the quantum computer may vary over time34296P-EP during the simulation of the quantum network protocol. Thus, an encoding used for representing the quantum state of the quantum network at a time T1 may be different from an encoding used for representing the quantum state of the quantum network at a different time T2. For example, a group of quantum constituents of the quantum computer may initially encode a first quantum system that arises in the quantum network at the time T1, and the same group of quantum constituents may thereafter encode a second, different quantum system that arises in the quantum network at the time T2.

[0093] It may be the case that the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. The above discussion, relating to encodings of local quantum systems of the quantum network, applies analogously to encodings of quantum information carriers that may be transmitted between nodes of the quantum network. At least one quantum information carrier that is transmitted in a quantum communication operation of the quantum network protocol may be encoded into one or more quantum constituents of the quantum computer. Similar to what was described above for local quantum systems of the quantum network, an encoding of a quantum information carrier into one or more quantum constituents of the quantum computer can be understood as a mapping which maps the quantum information carrier, or at least certain properties of the quantum information carrier, to the one or more quantum constituents. The encoding may be a partial encoding or a complete encoding as described herein. The quantum information carrier and the corresponding one or more quantum constituents may be quantum systems of a same type or a different type. Due to the encoding, the one or more quantum constituents may behave like the quantum information carrier, at least as regards one or more properties of the quantum information carrier that are relevant for the quantum network protocol. Information contained in the quantum information carrier may be mapped, via the encoding, onto the one or more quantum constituents. An operation performed on the quantum information carrier during the quantum network protocol may be simulated, via the encoding, by performing a corresponding operation (or sequence of operations) on the one or more quantum constituents.

[0094] It may be the case that a quantum information carrier is present during only a portion of the quantum network protocol (e.g. if the quantum information carrier is at some point measured and thereafter no longer needed in the quantum network protocol). Correspondingly, it may the34296P-EP case that the quantum information carrier is encoded into one or more quantum constituents of the quantum computer during only a portion of the simulation of the quantum network protocol.

[0095] At a time T during the quantum network protocol, one or more local quantum systems may be disposed at one or more respective nodes of the quantum network. The one or more local quantum systems together with one or more quantum information carriers transmitted between nodes of the quantum network as part of a quantum communication operation may be in a first quantum state at the time T. At a time T’ during the quantum simulation of the quantum network protocol, the one or more local quantum systems and the one or more quantum information carriers may be encoded into at least a portion of the quantum constituents of the quantum computer. At the time T’, at least a portion of the quantum constituents may be in a second quantum state that encodes at least one property of the first quantum state. In some cases, the entire first quantum state may be encoded into the second quantum state (complete encoding). In other cases, only a subset of properties of the first quantum state may be encoded into the second quantum state (partial encoding).

[0096] According to embodiments, in response to receiving the specification of the quantum network protocol as an input, a simulation of the quantum network protocol is performed using the quantum computer. The simulation is a quantum simulation. The quantum simulation of the quantum network protocol is a quantum mechanical process that includes evolving at least one quantum constituent of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol. A quantum simulation is distinguished from a classical simulation, which is a simulation using classical information processing only, i.e. without use of a quantum computer or other quantum information processing device.

[0097] A quantum simulation of the quantum network protocol can be understood as an evolution of the quantum constituents of the quantum computer that corresponds to the evolution of the quantum network caused at least by the sequence of operations of the quantum network protocol. The correspondence between the two evolutions may be established by virtue of the encoding which is being used. The encoding may be any encoding as described herein, particularly a complete or partial encoding, a local or global encoding, an encoding that varies over time, and the like. Generally, if an encoding establishes a correspondence between (properties of) the quantum states of the quantum network and (properties of) the quantum states34296P-EP of the quantum computer, said correspondence between the respective quantum states may imply, or induce, a correspondence between the quantum evolutions, more particularly a correspondence between a quantum evolution of the quantum network protocol and a quantum evolution of the quantum constituents of the quantum computer. For instance, continuing the example of a local encoding of a local quantum system of a quantum network into a subsystem of the quantum computer as described above, a quantum state= a |0> + b |1> + ... of the local quantum system of the quantum network may be encoded as a quantum state |^’> = a |^0> + b |^1> + .... of the associated subsystem of the quantum computer. Any quantum evolution (e.g. a unitary operation, measurement, completely positive map, and the like) of the quantum state is thus mapped to a corresponding quantum evolution of the quantum state |^’>, induced by the correspondence |0>.... Again, this is merely a possible example, and the disclosure shall not be limited thereto.

[0098] The quantum simulation of the quantum network protocol includes a simulation of the sequence of operations of the quantum network protocol. By virtue of the encoding, any quantum operation O (e.g. a local quantum evolution operation or a quantum communication operation) that is part of the sequence of operations of the quantum network protocol may be mapped to a corresponding quantum operation O’ acting on the quantum constituents of the quantum computer (possibly assisted by classical side-processing by the classical computing system). Performing the operation O’ on the quantum computer may provide a quantum simulation of the operation O. For example, a local evolution operation that is a unitary operation acting on a local quantum system may correspond, via the encoding, to a unitary operation acting on the constituent(s) in which the local quantum system is encoded. The same applies to e.g. a local quantum evolution operation that is a measurement, or more generally any completely positive map. Further, a quantum communication operation may involve an evolution of the quantum information carriers(s) that is / are being transmitted (e.g. due to the effects of noise acting on the quantum information carrier(s) during the transmission), which may again be represented by a completely positive map, which in turn corresponds to a completely positive map acting on the quantum constituents in which the quantum information carrier is encoded. The property that a quantum information carrier is transmitted from one node to another node can be simulated, i.e. that the quantum information carrier moves from a first node to a second node, can be simulated in various way, e.g. by performing a swap34296P-EP operation or by simply tracking classically (e.g. by the classical computing system 900) to which node the quantum information carrier belongs.

[0099] It may be the case that the aforementioned operation O is mapped, not to a single operation O’, but to a sequence of several operations O1’ O2’ ... acting on the quantum constituents, in which case it is said sequence which may provide a quantum simulation of the operation O. As a further alternative, several operations O1, O2, ... performed during the quantum network protocol may be grouped together to yield one effective operation O, which may then be mapped to a corresponding operation O’, or a sequence O1’ O2’ ..., acting on the quantum constituents, to provide a quantum simulation of the effective operation O (“event- based simulation”).

[0100] A classical communication operation from a first node to a second node may be simulated, for example, by recording classically (e.g. by the classical computing system 900) that certain classical data has been transmitted from the first node to the second node. Alternatively, this information may also be encoded in corresponding quantum constituents (e.g. quantum constituents that are provided in a computational basis state encoding the information under consideration).

[0101] A joint quantum state between component quantum systems disposed at respective nodes of the quantum network (e.g. entangled quantum state 14cde in Fig.1) may be simulated by likewise preparing the quantum constituents that correspond, via the encoding, to the component quantum systems in question in a joint quantum state. This may be achieved, for example, by applying an entangling operation (e.g. a unitary operation) acting jointly on said quantum constituents, as illustrated e.g. operation 910cde in Fig.9.

[0102] Fig.10 provides a further illustration of the quantum network protocol shown in Figs. 1-6. In Fig. 10, the quantum network protocol is schematically depicted as a sequence of operations 20a, 20c, 32ab, 40ce, 50b and 52df. For ease of presentation, the quantum systems 12a-f are not shown in Fig.10, yet these quantum systems are considered to be present at the respective nodes in the same manner as in Figs.1-6.

[0103] Performing a simulation of the quantum network protocol may include performing a simulation of a first portion of the quantum network protocol. The first portion may extend over a first time period from a first time to a second time. The first portion of the quantum network34296P-EP protocol may involve a first node set including one or more nodes of the quantum network. For example, in Fig.10, a first portion 1010 of the quantum network is indicated. In this example, the first portion 1010 involves the nodes 10a and 10b, which form the first node set in the present example. The first portion 1010 includes the operation 32ab (quantum communication operation). The first portion 1010 starts directly after the operation 20a has ended, and ends directly before the operation 50b starts, thereby defining the first time period for the present example.

[0104] In one example (as e.g. illustrated in Fig.10), the first portion of the quantum network protocol may include one or more operations performed during the first time period. Each of the one or more operations may be performed with respect to at least one node in the first node set. A single operation or multiple operations may be performed. For example, one or more local quantum evolution operations each acting on a node in the first node set, one or more classical communication operations each being performed between two nodes in the first node set, one or more quantum communication operations each being performed between two nodes in the first node set, or any combination thereof, may be performed during the first portion of the quantum network protocol.

[0105] In another example, it may be the case that, within the first portion of the quantum network protocol, no operations, i.e. no active operations, are performed with respect to the first node set as part of the quantum network protocol. Yet, noise may act on at least one node in the first node set during the first portion, i.e. during the first time period. In other words, a passive quantum evolution of the quantum system(s) corresponding to the first node set may take place during the first time period, even if no operations are actively performed. This is illustrated by the first portion 1010 shown in Fig.11. Said first portion 1010 corresponds to the evolution of the nodes 10a and 10b (= first node set) from the time directly after the operation 20a has ended until the time directly before the operation 32ab starts. No operations are actively performed during the first portion 1010 in Fig.11, yet the quantum systems under consideration may still be considered to evolve due to the presence of noise.

[0106] Further, both of the above cases may be combined, i.e. the first portion of the quantum network protocol may include one or more operations as described above, wherein the operation(s) in question, or a subset thereof, may be subject to noise. For example, also in the first portion in Fig.10, noise may be present.34296P-EP

[0107] The totality of the local quantum system(s) (if any) and the quantum information carrier(s) (if any) that are associated with the first node set during the first portion of the quantum network protocol may be considered to form a first quantum system. At the first time, the first quantum system may be in a first quantum state. During the first time period, the first quantum system may evolve from the first quantum state at the first time to a second quantum state at the second time, wherein the evolution may be caused by the operation(s) (if any) performed during the first portion of the quantum network protocol and noise (if any) acting on the first quantum system. The first quantum state and the second quantum state may be pure or mixed quantum states. In the example shown in Fig. 10, the first quantum system would be formed by the three quantum systems 12a and 12b.

[0108] The evolution of the first quantum system from the first quantum state to the second quantum state may be represented by an effective completely positive map. A completely positive map is a linear map that transforms each quantum state (pure or mixed) in another quantum state (pure or mixed). For example, any unitary operation is a completely positive map. Yet, there are also completely positive maps that are non-unitary.

[0109] The effective completely positive map may thus represent, for example, (a) an evolution of the first quantum system caused by a single operation performed during the first portion of the quantum network protocol, where the single operation may or may not be noisy, (b) an evolution of the first quantum system caused by multiple operations performed during the first portion of the quantum network protocol, where one or more of the operations in question may or may not be noisy, and (c) an evolution of the first quantum system caused merely by the presence of noise, without any operation being actively performed during the first portion of the quantum network protocol.

[0110] The effective completely positive map may be determined by a classical algorithm or a quantum algorithm. The effective completely positive map may be determined based on the specification of the quantum network protocol that is provided as an input and / or based on information regarding the noise (if any), e.g. a noise parameter, noise type, or full noise model. To determine the effective completely positive map, a respective completely positive map corresponding to each individual quantum operation (e.g. a local quantum evolution operation or a quantum communication operation) that is part of the first portion of the quantum network protocol may be determined. If a quantum operation is noisy, then a completely positive map34296P-EP representing the noise can be determined. The latter may either be a separate completely positive map representing the noise alone, or a completely positive map representing the action of the quantum operation and the noise jointly. If only noise is present during the first portion, a completely positive map representing the noise may be determined. Once individual completely positive maps are determined for respective parts of the first portion of the quantum network protocol, these individual completely positive maps may be composed with each other to yield an effective completely positive map corresponding to the first portion. In other examples, rather than determining the effective completely positive map as a composition of individual completely positive maps corresponding to smaller building blocks of the first portion, it may be more beneficial to determine the effective completely positive map directly, i.e. without breaking up the first portion into smaller parts.

[0111] In an example, the effective completely positive map may be determined by a classical algorithm, which may be executed on the classical computing system. In another example, the effective completely positive map may be determined by a quantum algorithm, which may be executed by the quantum computer. A quantum algorithm can be understood as a quantum computation carried out by a quantum computing system and is distinguished from a classical algorithm, which involves classical information processing only.

[0112] After the effective completely positive map corresponding to the first portion of the quantum network protocol has been determined, a quantum simulation of the effective completely positive map may be performed on the quantum computer, particularly by a quantum evolution unit thereof. As described above, by virtue of the encoding of the local quantum system(s) and / or quantum information carrier(s) that are present during the first portion (forming the above-described first quantum system) into one or more quantum constituents of the quantum computer, the effective completely positive map may be mapped to a corresponding completely positive map acting on the one or more quantum constituents of the quantum computer. The latter completely positive map may be performed on the quantum computer, thereby providing a quantum simulation of the effective completely positive map. For example, a completely positive map may be implemented on the quantum computer by using that any completely positive map may be performed as a unitary operation followed by a partial trace (Stinespring dilation). The implementation of completely positive maps (also called quantum channels) on a quantum computer is discussed in more detail below (see “Further aspects”, sections III-V).34296P-EP

[0113] Performing the simulation of the quantum network protocol may include determining an effective completely positive map representing an evolution of a first quantum system from a first time to a second time. The first quantum system may include one or more local quantum systems disposed at one or more respective nodes of the quantum network and / or one or more quantum information carriers transmitted between nodes of the quantum network. The evolution of the first quantum system from the first time to the second time may at least be caused by noise acting on the first quantum system or by one or more operations of the sequence of operations of the quantum network protocol or by a combination of said noise and said one or more operations. Performing the simulation of the quantum network protocol may include evolving, e.g. using a quantum evolution unit, at least one quantum constituent of the quantum computer to simulate the effective completely positive map.

[0114] The effective completely positive map may correspond to a first portion of the quantum network protocol. The quantum network protocol may include a plurality of portions. For each of the plurality of portions, a corresponding completely positive map may be determined analogous to the completely positive map corresponding to the first portion, and a simulation of the respective plurality of completely positive maps may be performed on the quantum computer.

[0115] The effective completely positive map may represent an evolution of the first quantum system caused by at least two operations of the sequence of operations of the quantum network protocol. It may be the case that each of the at least two operations is selected from the set consisting of a local quantum evolution operation, a classical communication operation and a quantum communication operation, particularly the set consisting of a local quantum evolution operation and a quantum communication operation.

[0116] The effective completely positive map may be non-unitary. An effective completely positive map that is non-unitary may arise, for example, where the quantum evolution of the first quantum system is caused at least in part by the presence of noise, or where the quantum evolution of the first quantum system is caused at least in part by measurement.

[0117] In some embodiments, the quantum network may be a noiseless quantum network, which may also be called an ideal quantum network. Accordingly, the quantum computer may perform a simulation of a noiseless quantum network protocol.34296P-EP

[0118] In other embodiments, the quantum network may be a noisy quantum network. A noisy quantum network can be understood as a quantum network where noise acts on one or more quantum systems of the quantum network (e.g. one or more local quantum systems and / or one or more quantum information carriers) during at least a portion of the quantum network protocol. Noise, also called decoherence, may be understood as an undesired, and in particular uncontrolled, interaction between a quantum system and the environment. Noise may cause a quantum evolution of the quantum system. Noise may transform a pure quantum state (coherent superposition) into a mixed quantum state. Noise is mathematically represented by a non- unitary completely positive map acting on the quantum system. There are different possible types of noise that may act on a quantum system (irrespective of whether this regards a quantum system of the quantum network or a quantum constituent of the quantum computer), such as depolarizing noise, dephasing noise, bit flip noise, amplitude damping noise, and the like. The present disclosure is not limited to any particular kind of noise.

[0119] The first portion of the quantum network protocol described above (e.g. first portion 1010 in Fig.10 or first portion 1010 in Fig.11) may be a first noisy portion. Noise may act on one or more quantum systems of the quantum network (e.g. one or more local quantum systems and / or one or more quantum information carriers) within the first noisy portion of the quantum network protocol. That the first portion is noisy does not necessarily imply that all operations performed during the first portion are subject to noise, but only that noise is present at least at some point during the first portion. In one example, the first noisy portion may include at least one quantum operation, such as a local quantum evolution operation or a quantum communication operation, that is subject to noise (for example, in Fig.10, operation 32ab may be affected by noise). In another example, it may be case that noise is present even though no operation is actively performed (e.g., noise may be present during first portion 1010 in Fig.11). The mere fact of doing nothing for a certain period of time (idling) may result in an evolution of the quantum system under consideration (first quantum system as described herein) due to the noise. The presence of noise, be it a noisy quantum operation or noise alone, will cause the quantum system to evolve according to a completely positive map, particularly a non-unitary completely positive map.

[0120] The noise occurring during the first noisy portion (and more generally the noise occurring at any point during the quantum network protocol) may be target noise. The term target noise can be understood as noise that is considered to be a property of the quantum34296P-EP network protocol that is to be simulated by the quantum computer. Particularly, a specification of the target noise (including e.g. a noise parameter and / or a noise model) may be provided as an input to the quantum simulation method, e.g. in addition to, or as a part of, the specification of the quantum network protocol. The quantum simulation performed by the quantum computer may simulate the quantum network protocol including a simulation of the target noise. In other words, the target noise itself (either standalone noise or noise that is part of a noisy quantum operation) may be simulated by the quantum computer. As described above, noise or noisy quantum operations, like any quantum evolutions, may be represented by a completely positive map, and completely positive maps may be simulated by the quantum computer as described herein.

[0121] The quantum network may be a noisy quantum network. The quantum network protocol may include a first noisy portion including target noise. Performing the simulation of the quantum network protocol may include simulating, using the quantum computer, the first noisy portion of the quantum network protocol. A simulation of the first noisy portion including a simulation of the target noise may be performed. The first noisy portion may include a noisy quantum operation, wherein the noisy quantum operation may be a quantum operation subject to the target noise. For example, the quantum operation may be a local quantum evolution operation or a quantum communication operation. Alternatively or additionally, the target noise may be standalone noise that is part of the first noisy portion of the quantum network.

[0122] Simulating the first noisy portion of the quantum network protocol may include determining information regarding the target noise. Particularly, the information regarding the target noise may include a noise parameter of the target noise or a noise parameter of a noisy quantum operation subject to the target noise. The first noisy portion of the quantum network protocol may be simulated based on the information regarding the target noise.

[0123] The information regarding the target noise may include a noise parameter. A noise parameter may indicate a strength of the target noise. A noise parameter may be a noise parameter of the target noise, e.g. as part of a noise model describing the target noise alone. Alternatively, a noise parameter may be a noise parameter of a noisy quantum operation subjected to the target noise. For example, in some cases it may not be possible to determine a noise model of the target noise separately, i.e. it might not be possible to clearly separate the34296P-EP target noise from a quantum operation affected by the target noise. In such cases, the noise parameter may be a noise parameter of the noisy quantum operation as a whole.

[0124] Determining information regarding the target noise, particularly a noise parameter, may include reading the information (e.g. from a memory), receiving the information (e.g. in a transmission of data), calculating the information, and the like. The information may be determined by a receiving unit that may be part of an apparatus as described herein. The receiving unit may include a classical computing system, a receiver, and the like, for determining the information.

[0125] Noise models as well as examples of different types of noise and associated noise parameters are discussed in more detail below (see “Further aspects”, section III.B).

[0126] In any realistic quantum computer, noise is always present at least in small amounts. Quantum constituents, and quantum operations performed on quantum constituents, are subject to at least small amounts of noise. Noise affecting a quantum computation performed by the quantum computer will be referred to herein as source noise. Source noise is a physical, real- world phenomenon occurring in a physical quantum system, namely the quantum computer. Which type of source noise is present depends on the physical circumstances, such as the particular type of the quantum constituents that are being used, and the particular type of apparatus being used to interact with the quantum constituents. Source noise present during a quantum computation performed by the quantum computer is distinguished from target noise acting on the quantum network. Target noise is part of the process to be simulated, whereas source noise is physically present in the quantum computer, i.e. the system performing the quantum simulation.

[0127] In conventional quantum computations / simulations, source noise is considered detrimental for the performance of the quantum computer. Accordingly, normally one would aim to suppress the source noise (e.g. by performing quantum error-correction), thus striving to perform a quantum computation where the source noise is small. In other words, a conventional aim is to attempt to perform a quantum computation or simulation that approximates an ideal, i.e. noiseless, quantum computation / simulation, by aiming to reduce the noise.

[0128] Embodiments described herein differ from the above conventional approach. As described above, the quantum simulation method of the present disclosure may include34296P-EP performing a simulation of target noise that is present during the quantum network protocol. For simulating the target noise, the source noise present in the quantum computer is not considered as a disadvantage that shall be suppressed in order to strive towards an ideal quantum computation, but rather as a part of the quantum computational process that is leveraged in order to perform - and improve - the simulation of the target noise. Rather than attempting to reduce the source noise, the source noise is transformed, or re-shaped, to match the target noise, by performing a suitable sequence of noisy quantum operations (i.e. quantum operations subject to the source noise) on the quantum computer. The inventors have found that, if the target noise in the quantum network is relatively strong - which is the realistic scenario in many current applications - there is no need to try and suppress the source noise. Rather, the quantum operations performed by the quantum computer according to embodiments described herein aim to transform the source noise, without necessarily reducing the strength of the source noise, or by even increasing said strength if needed, so that the type and strength of the transformed source noise matches, i.e. simulates, the target noise.

[0129] Fig. 12 illustrates a noisy sequence 1200 of quantum operations 1231 through 1235. One or more of the quantum operations 1231 through 1235 may include source noise. Additionally, or alternatively, source noise may be present at certain times (or at all times) during the noisy sequence even when no operation is being performed, i.e. when the quantum computer is idling. The noisy sequence 1200 is an illustrative example of a noisy sequence of quantum operations that may simulate the first portion 1010 shown in Fig.10, in the case where said first portion 1010 is affected by target noise (in which case the first portion 1010 is referred to as a noisy first portion 1010). In the example under consideration, quantum operations 1231 and 1234 may act on subsystem 80a, quantum operations 1232 and 1235 may act on subsystem 80b, and quantum operation 1233 may act jointly on subsystems 80a and 80b. The quantum operations in question may further act on ancillary constituents of the quantum computer, which are not shown in Fig.12. It shall be understood that the noisy sequence 1200 is merely an example for the purpose of illustrating the concepts in question.

[0130] The noisy sequence 1200 may provide a quantum simulation of the noisy first portion 1010. The quantum operations 1231 through 1235 may be determined by performing an optimization algorithm. The quantum operations 1231 through 1235 may be selected such that the noisy sequence 1200 including the source noise that is present during the noisy sequence 1200 provides a suitable simulation of the noisy first portion 1010. The source noise occurring34296P-EP during the noisy sequence 1200 may be transformed, or re-shaped, to obtain a simulation of the noisy first portion 1010. In particular, it may be the case that an ideal sequence of quantum operations obtained by replacing each of the quantum operations 1231 through 1235 by a noiseless version thereof provides a less accurate simulation than the noisy sequence 1200. The presence of the source noise in the noisy sequence 1200 may provide an improved simulation of the noisy first portion 1010 as compared to a noiseless sequence.

[0131] Simulating the first noisy portion (e.g. noisy portion 1010) of the quantum network protocol may include performing a first noisy sequence of quantum operations (e.g. noisy sequence 1200) on one or more quantum constituents of the quantum computer. Source noise may be present during at least a portion of the first noisy sequence of quantum operations. The first noisy sequence of quantum operations may simulate the first noisy portion of the quantum network protocol. The first noisy sequence may be performed using a quantum evolution unit as described herein. That source noise is present during at least a portion of the first noisy sequence may include that at least one quantum operation in the first noisy sequence is subject to source noise and / or that standalone noise is present during at least a portion of the first noisy sequence.

[0132] The first noisy sequence of quantum operations including the source noise may simulate the first noisy portion of the quantum network protocol. The first noisy sequence of quantum operations may be configured to transform source noise present during the first noisy sequence to target noise present during the first noisy portion of the quantum network protocol. The first noisy sequence may provide a more accurate simulation of the first noisy portion than a noiseless version of the first noisy sequence. The noiseless version may be a sequence of quantum operations obtained by removing the source noise from the first noisy sequence.

[0133] The source noise may be different from the target noise. A type and / or a strength of the source noise may be different from a type and / or a strength of the target noise. The type of noise may refer to the particular noise model that applies to the noise in question, for example whether the noise is depolarizing noise, dephasing noise, amplitude damping noise, or the like. The strength of the noise may refer to the value(s) of one or more noise parameters of the noise, wherein a larger value of a noise parameter may correspond to a larger strength of the noise. Embodiments described herein, which transform, or re-shape, the source noise to obtain a quantum simulation of the target noise (or a noisy portion of the quantum network including34296P-EP target noise), may involve source noise that is different in type and / or strength than the target noise.

[0134] In order to determine an optimal noisy sequence of quantum operations simulating the noisy portion under consideration of the quantum network protocol, information regarding the source noise may be used. This is different from other approaches, where an optimal sequence of quantum operations to simulate the noisy portion is first determined in the absence of source noise, i.e. in an ideal scenario - and thus ignoring the source noise altogether - and where the effects of the source noise on the simulation are only considered thereafter. In such approaches, it is normally the case that the addition of source noise deteriorates the quality of the simulation, i.e. the ideal sequence of operations would provide a more accurate simulation than the noise version. As described above, in embodiments described herein, the converse is true.

[0135] Information regarding the source noise may include one or more noise parameters, or more specifically a full noise model, of the source noise. For example, one or more noise parameters of the source noise may be used as input data to an optimization algorithm, together with other input data, such as information regarding the target noise to be simulated. The optimization algorithm may perform an optimization over a set of quantum operations (e.g. a particular set of quantum operations that can realistically be executed by the quantum computer at hand) to find a noisy sequence of quantum operations (i.e. a sequence in which the source noise occurs) that provides an optimal simulation of the noisy portion of the quantum network protocol. For example, the optimization may aim to minimize a distance (e.g. channel infidelity) between the noisy sequence of quantum operations and the target noise. The optimization algorithm itself (or more generally any other algorithm to determine the sequence of quantum operations) may be a classical algorithm or a quantum algorithm, in either case having information regarding the source noise as classical input data. Further technical aspects are described below (see “Further aspects”, sections V.A and V.B).

[0136] Instead of using information regarding the source noise, or in addition thereto, in order to determine a suitable (optimal) noisy sequence of quantum operations simulating the noisy portion of the quantum network protocol, a quantum algorithm may be performed by the quantum computer that involves a physical manifestation of the source noise. Since the quantum computer is subject to the source noise by the very physical set-up of the quantum computer, the source noise will be present as a part of any quantum algorithm ran on the quantum34296P-EP computer. Within this paradigm, a noisy sequence of quantum operations simulating the noisy portion of the quantum network protocol may be determined by performing, for example, an optimization algorithm similar to what was described above, yet in the present case the optimization algorithm is limited to be a quantum algorithm and, rather than having information regarding the source noise as classical input data, the source noise is physically present within the optimization algorithm as a quantum process. Thus, also in this case, the source noise is an inherent, deliberate part of the determination of the noisy sequence of quantum operations that optimally simulates the noisy portion of the quantum network protocol. This approach may be beneficial, for example, in cases where information regarding the source noise is not sufficiently available. Further technical aspects are described below (see “Further aspects”, section V.C).

[0137] The first noisy sequence of quantum operations may be determined using information regarding the source noise and / or using a physical manifestation of the source noise. The first noisy sequence of quantum operations may be determined by a classical or quantum algorithm using information regarding the source noise and / or by a quantum algorithm performed on the quantum computer while the quantum computer is affected by the source noise during at least a portion of the quantum algorithm. That the quantum computer is affected by the source noise can include that one or more of the quantum constituents and / or one or more quantum operations performed by the quantum computer are subject to the source noise.

[0138] The first noisy sequence of quantum operations may be determined by performing an optimization algorithm, particularly an optimization algorithm to minimize a distance between the first noisy sequence of quantum operations and the first noisy portion of the quantum network.

[0139] The optimization algorithm may be a classical algorithm or a quantum algorithm. The optimization algorithm may use information regarding the target noise to determine the first noisy sequence of quantum operations. The information regarding the target noise may include a noise parameter of the target noise or of a quantum operation affected by the target noise.

[0140] The optimization algorithm may include a classical or quantum algorithm that uses information regarding the source noise to determine the first noisy sequence of quantum operations. Additionally or alternatively, the optimization algorithm may include a quantum algorithm performed on the quantum computer while the quantum computer is affected by the source noise during at least a portion of the quantum algorithm. The optimization algorithm34296P-EP may determine the first noisy sequence of quantum operations as an optimal sequence of quantum operations simulating the first noisy portion of the quantum network protocol. The optimal sequence may depend on the source noise.

[0141] Performing the simulation of the quantum network protocol may include measuring at least one quantum constituent to obtain a readout. The readout may include information regarding a property of the quantum network protocol. A quantum evolution unit of the apparatus as described herein may include a measurement device for measuring one or more quantum constituents. The measurement device may be configured for measuring at least one quantum constituent to obtain the readout.

[0142] Additionally or alternatively, performing the simulation of the quantum network protocol may include preparing a second quantum state of at least one quantum constituent of the quantum computer. The quantum state may be an output quantum state of the simulation. The second quantum state may correspond to a first quantum state prepared during the quantum network protocol (e.g. at the end of the quantum network protocol or at an intermediate time during the quantum network protocol). The at least one quantum constituent prepared in the second quantum state may be transmitted to a receiver. Said transmission may be a physical transmission of the at least one quantum constituent to the receiver, i.e. a quantum communication operation. For example, the receiver may further process the second quantum state, e.g. perform a measurement of the second quantum state to obtain a readout, and determine a property of the quantum network based on the readout.

[0143] The quantum computer may be a programmable quantum computer configured for quantum simulation of a plurality of different quantum networks. A programmable quantum computer may include a quantum computer that has the flexibility to receive and process a plurality of different inputs (in the present case, specifications of a plurality of different quantum network protocols). A programmable quantum computer may be such that the quantum computations that can be performed by the quantum computer are not restricted to one particular fixed quantum computation (or a fixed small set of quantum computations) that is e.g. hard-wired in the system. An example of a programmable quantum computer is a universal quantum computer, which is capable of performing arbitrary quantum computations. Yet not every programmable quantum computer needs to be universal.34296P-EP

[0144] The plurality of different quantum networks that a programmable quantum computer may be configured to simulate may include: quantum networks having a different total number of nodes; quantum networks where the quantum network protocols have different kinds and / or a different number of operations (local quantum evolution operations, classical or quantum evolution operations); quantum networks where the properties of the quantum systems that are present in the quantum network are different (e.g. whether the quantum systems are qubits, d- level systems, or the like); quantum networks where the physical implementation of the quantum systems that are present in the quantum network are different; quantum networks where the target noise is different; or any combination thereof. Embodiments described herein are particularly distinguished from quantum simulations where a fixed, non-programmable physical quantum system is used to simulate one particular quantum network or a particular small set of quantum networks.

[0145] According to a further embodiment, a method of determining a control layout for a quantum computer is provided. The method includes receiving, as an input, a specification of a quantum network protocol associated with a quantum network. The quantum network is a distributed network including a plurality of nodes, wherein a local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local quantum evolution operations, wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination thereof, is provided. Therein, according to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations, wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. Further, according to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. Further, according to (c), a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum34296P-EP systems, wherein K is at least two. The method includes, in response to the receiving, determining a control layout for the quantum computer for performing a simulation of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol. The method may be a computer-implemented method, which may, for example, be carried out by a classical computing system. The method may include any features, either alone or in combination, described in relation to a quantum simulation method according to embodiments described herein.

[0146] A control layout for a quantum computer can include information that is to be transmitted to a controller of a quantum computer. The control layout can include control instructions for the quantum computer or information that allows to determine control instructions therefrom. The control instructions may cause the quantum computer to perform a quantum simulation of the quantum network protocol as described herein.

[0147] A control layout can be a control layout of an entire quantum computation from start (preparation of initial quantum state) to finish (measurement(s) to provide read-out), or can alternatively be a control layout for a portion of a quantum computation.

[0148] Determining the control layout for the quantum computer may include determining an encoding of local quantum systems of the quantum network into at least a subset of the quantum constituents of the quantum computer. Each of the local quantum systems may be disposed at a respective node of the quantum network during at least a portion of the quantum network protocol. According to the encoding, each of the local quantum systems may be encoded into at least one quantum constituent of the quantum computer. Determining the control layout for the quantum computer may include determining whether and how an encoding shall be modified during the quantum simulation of the quantum network protocol.

[0149] Determining the control layout for the quantum computer may include determining one or more quantum operations to be performed by the quantum computer and / or one or more classical operations to be performed by a classical computing system connected to the quantum computer for performing a simulation of the quantum network protocol.34296P-EP

[0150] Determining the control layout for the quantum computer may include determining a first noisy sequence of quantum operations configured to act on one or more quantum constituents of the quantum computer. The first noisy sequence may be configured to simulate a first noisy portion of the quantum network protocol.

[0151] Determining the control layout for the quantum computer may include performing an optimization algorithm to determine the first noisy sequence of quantum operations.

[0152] According to a further embodiment, a control layout for a quantum computer for performing a simulation of a quantum network protocol associated with a quantum network is provided. The quantum network is a distributed network including a plurality of nodes. A local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local quantum evolution operations, wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination thereof, is provided. Therein, according to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations, wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. Further, according to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. Further, according to (c), a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum systems, wherein K is at least two. The control layout includes control instructions for the quantum computer, or information that allows determining said control instructions. The control instructions cause the quantum computer to perform a simulation of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.34296P-EP

[0153] The control layout may include an encoding of local quantum systems of the quantum network into at least some quantum constituents of the quantum computer, or information that allows determining said encoding. Each of the local quantum systems may be disposed at a respective node of the quantum network during at least a portion of the quantum network protocol. According to the encoding, each of the local quantum systems may be encoded into at least one quantum constituent of the quantum computer.

[0154] The control layout may include a control layout for one or more quantum operations to be performed by the quantum computer and / or a description of one or more classical operations to be performed by a classical computing system connected to the quantum computer for performing the simulation of the quantum network protocol.

[0155] The control layout may include a control layout for a first noisy sequence of quantum operations configured to act on one or more quantum constituents of the quantum computer. The first noisy sequence may be configured to simulate a first noisy portion of the quantum network protocol.

[0156] According to a further embodiment, a data carrier or data carrier signal carrying information representing the control layout according to embodiments described herein is provided.

[0157] Fig.13 shows an apparatus 1300 for performing a quantum simulation.

[0158] According to a further embodiment, an apparatus 1300 for performing a quantum simulation is provided. The apparatus 1300 includes a quantum computer 700. The quantum computer 700 includes quantum constituents 70 and a quantum evolution unit 1310 configured to perform an evolution of at least some of the quantum constituents 70. The apparatus 1300 includes a classical computing system 900. The classical computing system 900 is configured for receiving, as an input, a specification 902 of a quantum network protocol associated with a quantum network, wherein the quantum network is a distributed network comprising a plurality of nodes, wherein a local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol. The quantum network protocol includes a sequence of operations. The sequence of operations includes one or more local quantum evolution operations, wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes. Further, (a), (b) or (c), or any combination34296P-EP thereof, is provided. Therein, according to (a), the sequence of operations of the quantum network protocol includes one or more classical communication operations, wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network. According to (b), the sequence of operations of the quantum network protocol includes one or more quantum communication operations, wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network. According to (c), a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state of K component quantum systems, wherein K is at least two. The classical computing system 900 is configured for, in response to the receiving, encoding local quantum systems of the quantum network into at least a subset of the quantum constituents 70 of the quantum computer 700, wherein each of the local quantum systems is disposed at a respective node of the quantum network during at least a portion of the quantum network protocol, wherein each of the local quantum systems is encoded into at least one quantum constituent 70. The quantum computer 700 is configured for performing a simulation of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some of the quantum constituents 70 from a first quantum state to a second quantum state using the quantum evolution unit 1310 to simulate the sequence of operations of the quantum network protocol.

[0159] The apparatus 1300 may be configured to perform any operation, or combination of operations, of a quantum simulation method according to embodiments described herein.

[0160] The quantum evolution unit 1310 may be configured to perform any kind of quantum evolution of the quantum constituents 70, including, but not limited to, unitary operations, measurements, adiabatic evolutions, dissipative evolutions, and the like. The quantum evolution unit 1310 may include subunits configured to perform dedicated types of operations, e.g. a unitary evolution unit to perform unitary operations, a measurement unit to perform measurements, and the like. The physical realization of the quantum evolution unit 1310 and its subunits may depend on the type of quantum constituents being used and the type of quantum evolution(s) being performed by the quantum evolution unit 1310. Illustrative examples of34296P-EP quantum evolution units 1310 include laser systems, voltage control units, optical components, controllable magnetic fields, charged-coupled device (CCD) cameras, photodetectors, etc.

[0161] A classical computing system 900, or classical computer, can be understood as a computing system that processes information using only classical information carriers, such as classical bits. The term “classical” can in this context be understood as “not quantum”. A classical computing system can include, for example, a personal computer or a network of personal computers. The classical computing system 900 may be configured to perform any classical information processing task or any combination of classical information processing tasks that are part of a quantum simulation method according to embodiments described herein.

[0162] The quantum computer 700, and particularly the quantum evolution unit 1310, may be connected to the classical computing system 900. Classical data may be transmitted between the quantum computer 700 and the classical computing system 900. The classical computing system 900 may be, or may be part of, a controller for controlling the quantum computer 700. The controller may instruct the quantum evolution unit 1310 to perform a quantum simulation of the quantum network protocol based on the specification 902. The controller may receive information from the quantum evolution unit 1310, e.g. outcome(s) of one or more measurements performed by the quantum evolution unit 1310. Further aspects I. Introduction

[0163] Quantum technologies utilize the features of quantum systems and offer the possibility for new and unexplored applications. Quantum computers promise to solve optimization and other problems in logistics, finances, chemistry and drug design that are not accessible with classical devices. Quantum simulators are discussed as tools to simulate physical models and provide insights into phenomena such as high-temperature superconductivity, and quantum sensors offer the possibility to measure physical quantities with unprecedented precision. Connecting such quantum devices to form a quantum network, or ultimately a quantum internet, leads to even more possibilities and unleashes the full power of quantum devices. Connecting quantum sensors to a quantum sensor network opens the way to measure spatially correlated quantities and applications such as high-resolution imaging. Connecting small-scale quantum computers makes them even more powerful, allowing access to the exponentially large Hilbert34296P-EP space. Planning and building such quantum networks is hence of importance not only to connect quantum devices and make them more powerful but also to make them broadly accessible.

[0164] However, the same features that are responsible for the power of connected quantum devices also pose the main hindrance to simulating these systems. The exponentially growing state space makes an exact and accurate description of devices and protocols very demanding, if not impossible. Nevertheless, classical approaches for simulating quantum networks have been developed and have been applied to simulate certain network devices and protocols classically. In order to cope with the inherent difficulties, the treated systems are either significantly small in size (at most a few tens of qubits) or very simplified models that are used to describe quantum states and noise processes. Techniques such as the stabilizer formalism allow one to classically simulate specific cases of interest efficiently, even for large systems, i.e., networks of multiple nodes. However, when dealing with general situations, in particular noise that is not described by a simple error model, the effort to perform an exact simulation scales exponentially with the number of systems, i.e., the number of nodes in the network, limiting the size of simulations based on classical approaches.

[0165] In the present disclosure, it is proposed to use a quantum computer, particularly a noisy intermediate-scale quantum (NISQ) device, to perform the simulation of quantum networks and their features, as illustrated in Fig.14. The left-hand side of Fig.14 shows a quantum network including a plurality of nodes as described herein. A quantum network protocol associated with the quantum network is simulated by a quantum computer according to the method described herein, as illustrated in the right-hand side of Fig.14. Any quantum network protocol, including noise therein, can be suitably simulated using the quantum computer, overcoming classical simulator limitations.

[0166] Using a quantum computer to simulate a quantum network protocol not only overcomes the problem of exponentially growing state space, but has the additional advantage that noise in the quantum computer (e.g. a NISQ device) does not pose a hindrance to successful simulation, but is rather a feature that is taken advantage of. Usually, in applications of quantum technologies, noise and imperfections are undesired and considered as a hindrance from harnessing the full power of quantum devices, and hence need to be suppressed and mitigated. In contrast, in the present disclosure it is an aim to simulate quantum systems that are themselves noisy, and their interaction via even noisier quantum channels - and it is exactly the34296P-EP performance of the quantum network taking such noise and imperfections into account that the quantum computer aims to simulate. The Ansatz adopted by the inventors is hence to modify and tailor the noise present in the quantum computer (source noise as described herein) in such a way that it represents the noise present in the quantum systems and operations to be simulated (target noise as described herein). Relevant error models are considered to describe noisy quantum systems and operations in the quantum network, such as sources, channels, and memories, including dephasing, depolarizing, and Pauli channels, but also amplitude damping, absorption or combinations thereof. To this aim, techniques and optimizations of how to simulate noise processes and gates taking imperfections into account are introduced - not by considering an idealized, perfect implementation and studying the effect of the source noise thereafter, but by actively reshaping the source noise to obtain a highly accurate approximation of the target noise for storage, gates, and channels. In many relevant cases, even exact modeling of the desired target noise model is possible, e.g., when dealing with Pauli noise channels in the NISQ devices with source noise that is smaller in strength than the target noise. Notice that target noise in channels and memories in quantum networks is typically much larger than source noise in quantum computers.

[0167] The approach taken by the inventors has multiple advantages as compared to classical simulation of quantum networks, and also compared to the direct benchmarking of existing networks. On the one hand, one can simulate larger systems, and use accurate or exact error models. On the other hand, one has the flexibility to easily change parameters and even network topologies, and perform feasibility studies, optimization of protocols and network structures. There is no latency due to probabilistic processes or classical communication times, and improved readout is possible since locality restrictions do not apply.

[0168] In other quantum simulators, well-controlled quantum systems may be used to simulate ground states or the dynamics of other physical models - e.g., strongly correlated systems or field theories. In contrast, embodiments described herein are not concerned with the simulation of physical Hamiltonians or Master equations, but with discrete processes and complex protocols. Furthermore, the nodes of a quantum network may themselves be complex quantum systems that include multiple qubits, and an aim is to obtain an accurate simulation of nodes, channels, and devices, and on the other hand in the complex interaction between them when executing a communication protocol.34296P-EP II. Quantum simulation of quantum networks and the advantages thereof II.A. Quantum simulation of quantum networks

[0169] The general approach for the simulation of quantum networks using quantum computers is described in the following. A quantum network includes multiple spatially separated nodes, where each node may hold quantum devices (local quantum systems) of different kinds. Network nodes may be connected to each other via classical channels (to transmit classical information), and / or by quantum channels to exchange quantum information. The respective geometries of the classical and quantum channels determine the network topology, and are in principle distinct and independent from each other. Operations on devices, processes, and the transmission of quantum information may be imperfect and noisy, and also the transmission of classical information or classical side-processing can lead to delays and errors that need to be taken into consideration.

[0170] A quantum network protocol may involve local quantum evolution operations (e.g., measurements, unitary operations, generation of photons, etc.) at respective nodes, and transmission of quantum and classical information between the nodes. In the quantum simulation of such a protocol, all devices (i.e. the local quantum systems) may be represented on a quantum computer. Local quantum systems are typically encoded into quantum states of one or several qubits (or other quantum constituents of the quantum computer), the quantum network protocol may be simulated by operations and measurements performed on these quantum states in a quantum circuit fashion. The effect of imperfect operations, transmissions and waiting times in the quantum network protocol may be simulated by performing the respective operations on the total quantum state stored in the quantum computer. The quantum computer may be configured for performing standard quantum operations, i.e., unitary operations from a universal gate set (e.g. arbitrary single-qubit operations and a two-qubit operation, e.g., a CNOT gate) and projective single-qubit measurements. The simulation can be done in different ways. A first possibility is to directly simulate the time evolution of the quantum network as it would occur in the quantum network protocol, at all times. A second, more economic possibility, is to perform an event-driven, or event-based, simulation. That is, the quantum state of the quantum computer is only updated if a relevant event occurs in the quantum network protocol, by performing the associated completely positive map that describes the effective evolution of the quantum network after this finite time. For instance,34296P-EP consider an entangled state shared between two nodes in the network that is stored for some time in a quantum memory, and further processed at some later time when a classical signal arrives. One can compute the effective completely positive map (CPM) that describes the evolution of the state of the quantum network after this time, and then only implements a simulation of this CPM on the quantum computer, without computing intermediate states, or actually waiting for this time. Quantum devices (local quantum systems).

[0171] The quantum simulation may include encoding each quantum device (source, memory, interface, etc.) of the quantum network – that is, each local quantum system – as a quantum state of one or more quantum constituents of the quantum computer. Typically, multiple qubits may be used to provide an accurate description or model of the quantum device. In some cases, additional ancillary quantum systems may be used to simulate specific features, or allow for an accurate, direct simulation of the underlying noise process.

[0172] Multiple quantum devices are represented by a tensor product of the corresponding quantum states, and the required number of qubits adds up. These states may become entangled when devices interact or exchange quantum information during the quantum network protocol.

[0173] The number of required qubits for the simulation may change dynamically. This can be due to adding or removing devices or nodes from the network, or the generation of additional quantum systems in the protocol (e.g., a photon that is generated and subsequently transmitted from one node to another node). Furthermore, measurement of quantum systems that thereafter no longer participate in the quantum network protocol reduces the size of the quantum state to be stored and processed by the quantum computer. For processes that only involve certain nodes or devices (which is a typical case), the total size of the required quantum state is only given by the devices that participate in the process. One may also simulate certain parts of a process using a quantum computer, which then requires fewer qubits. If different processes within the quantum network protocol are separated from each other, i.e., do not involve the same local quantum systems, one maintains a tensor product structure between them. This allows one to perform simulations separately or even sequentially, thereby reducing the required size of the simulating device.34296P-EP

[0174] The quantum systems that take part in a quantum network protocol as well as the quantum constituents of the quantum computer are not limited to qubits, and can be arbitrary quantum systems. Local quantum evolution operations

[0175] When a quantum network protocol is executed, quantum operations on the local quantum systems may be performed. Typically, this will include unitary operations to manipulate quantum states. Since in real-world applications all operations are imperfect and noisy, a simulation of a realistic quantum network shall account for the noise in the simulation. In an event-based simulation, one may apply the corresponding CPM that describes the noisy operation. To do so, a model based on the physics of the underlying noisy process may be used, from which the CPM can be determined. In the simulation, one or more quantum operations corresponding to the CPM are performed on the stored quantum state of the quantum computer. As described in more detail below, this involves sequences of gates and measurements performed on the quantum computer, and typically involves some ancillary systems.

[0176] Quantum network protocols may involve measurements, e.g., when performing entanglement purification, or when manipulating a multipartite entangled state to generate some other state on a subsystem. Similar to other operations, these measurements are noisy and imperfect in reality, and can be described by a positive operator-valued measure (POVM). The POVM can be implemented in the simulation using sequences of gates and (projective) measurements performed on an enlarged system including one or more quantum constituents. Quantum communication operations

[0177] In a quantum network, quantum information may be transmitted from one node or device to another. This is done by sending quantum information carriers, e.g. photons, atoms and the like. Transmitted photons (or other quantum information carriers) may suffer from loss, and due to interactions with environment, the transmission will be in general imperfect. This can again be described by a CPM, possibly acting on a higher dimensional system to include decay or loss. In the field of quantum communication, these CPMs are sometimes termed noise channels, and several models have been considered and analyzed. The ideal operation is the identity in this case, and the effect of noise and imperfections is again described by the CPM.34296P-EP Orchestration

[0178] The orchestration of the quantum network protocol, and the orchestration of the simulation thereof, may be done using a classical control plane, which steers classical and quantum devices, and classical and quantum channels. Events trigger actions, which can, e.g., be operations, measurements or transmission of classical information. The orchestration and execution of a quantum network protocol is done by exchanging classical messages, which trigger actions, including the transmission of quantum information or quantum operations performed on local quantum systems. In principle, the orchestration can also be performed via quantum states, which encode processes and actions, thereby steering the processes in a quantum way. In the simulation, this orchestration needs to be translated into an appropriate execution of CPMs and POVMs on the stored quantum state that describes network devices.

[0179] Notice that it is also possible to simulate only specific parts of the quantum network protocol using a quantum computer. In this case, the results of the quantum simulation may be integrated into a classical simulation of the remainder of the quantum network. In this way, resource-consuming parts of simulation can be performed on a quantum computer, while other parts can be simulated classically. II.B. Advantages II.B.1. Advantages over classical network simulation

[0180] In the following, advantages of quantum simulations of quantum network protocols are discussed as compared to classical simulations.

[0181] System size. In a quantum processor, one can treat large quantum systems, and hence, quantum networks with multiple nodes, which is not possible with classical simulation methods. One is thereby mainly restricted by the available size of the quantum computer, where only a few ancilla qubits may already suffice to simulate the action of different noise channels. This includes the treatment of multiplexing schemes with shared resources (e.g., memories), the distribution of large multipartite entangled states in a network, or protocols that operate on multiple copies, e.g., entanglement purification or state verification schemes. For entanglement- based quantum networks, where large resource states are locally manipulated to obtain desired target configurations to fulfill requests, the quantum simulation approach is also well suited. The error models that can be easily simulated by quantum simulation are much broader than34296P-EP the few simplified models where efficient classical descriptions exist, and in a quantum simulation even the exact treatment of noise maps is possible with a very small overhead of a few qubits.

[0182] Accurate error models. When utilizing quantum computers, one can use exact or very accurate error models in the simulation and is not restricted to (over)simplified models that might not cover relevant effects that are crucial to judging protocol performance and emerging behaviors of large networks. Using simple models is however necessary for classical simulations to be able to treat systems of several tens or even more than a hundred qubits. Example

[0183] We illustrate the advantages of a quantum simulation using a moderate-sized quantum computer. Consider a quantum network protocol where a multipartite entangled state, say a graph state or a Dicke state, should be distributed over a large distance between ^^ nodes. Noise in channels and noise in local devices is assumed to be general, and not restricted to be symmetric or only of Pauli form. Hence most existing classical simulation methods cannot be applied. Repeater protocols require operations on at least two copies of the state to perform entanglement swapping or state merging, as well as entanglement purification to increase fidelity, whereas more efficient schemes even operate on multiple copies (say ^^) in parallel.

[0184] For simplicity, we consider only a single ^^ → 1 entanglement purification step, i.e., theiteration operates on ^^ identical copies and outputs a single purified copy. The ^^ input states need to be first generated, and the output state stored while the processor restarts to generate a second purified copy. Thus, to proceed with the second iteration ^^ purified copies need to begenerated, meaning ^^(1) = 2(^^^^) qubits are required to simulate the process using a quantumcomputer. Therefore, ^^ steps of entanglement purification require ^^(^^) = 2^^(^^^^) qubits. Aclassical simulation method needs to operate jointly on ^^ copies of a ^^ qubit state, i.e., ondensity matrices of the size 2^^^^ × 2^^^^. Even for relatively small quantum networks, say a stateof ^^ = 10 qubits and operating on ^^ = 3 copies, this vastly exceeds the available memory andcomputational power, as 260 ≈ 1018. Multiple steps ^^ require the additional storage of theresulting state, but this overhead is insignificant as compared to the already required resources.In turn, a quantum computer of size ^^ = 30 already suffices to perform a single step in thepurification, with a linear growth in required resources when multiple steps are considered.34296P-EP II.B.2. Advantages over network benchmarking

[0185] Simulating networks rather than physically building and benchmarking them is also of advantage, as discussed in the following.

[0186] Feasibility studies. The physical realization of a quantum network in real life might be extremely costly and challenging - or even impossible with available technology and devices, and it might be desirable to perform a feasibility study using quantum simulations first. Thereby, the required quality of devices and channels can be determined - or it can be assessed if particular devices and resources suffice to achieve the desired functionality. Since there may be emerging effects in complex networks, it is beneficial to identify them beforehand and so that the network can be designed accordingly. Bottlenecks, memory requirements and other features can be identified in the simulation, and the influence of certain network elements or channels on them can be studied.

[0187] Parameter modifications. One can freely change parameters in the quantum simulation, and access parameter regimes that are not (yet) accessible with current technology. This allows one to investigate the effect of such improvements, e.g., of the capabilities of a specific device, and judge how large the effects on overall protocol and network performance are. In this way, one can determine in advance how future developments can improve network performance, and assess the necessary quality of components - and identify the most crucial elements where the largest overall improvement is possible. Similarly, one can simulate the effect of failing or badly performing devices.

[0188] Flexible topology. One can easily change the topology of a network in the quantum simulation, whereas this is very difficult and requires adding new channels (e.g., optical fibers) and building additional nodes in real-world physical networks. This is not only very costly but also time-consuming when using real devices. In turn, the quantum simulation opens the possibility to test and investigate different topologies. In a similar way, bottlenecks in a planned network can be identified, and different ways to avoid them can be tested.

[0189] Latency and waiting times. In real-world quantum networks, latency and waiting times -e.g., for classical signals to arrive after measurements, or to deal with probabilistic processes that need to be repeated until success - play an important role, and lead, e.g., to small communication and key rates for long-distance communication schemes based on quantum34296P-EP repeaters. These waiting times also occur when benchmarking a network, and testing or investigating a protocol and its performance. There are no such waiting times in the quantum simulation - the effect on quantum states when sending them through a noisy channel, or keeping them in a quantum memory for a certain time can directly be described by a completely positive map that is implemented on the quantum computer in one step. One only needs to take these effects into account in the simulation software, and not wait for processes to actually occur or classical signals to arrive. Qubits in the quantum processor used for simulation are close to each other, and communication times are much smaller than between nodes that may be separated by hundreds or thousands of kilometers.

[0190] Network and protocol optimization. One can use the quantum simulation to optimize networks, e.g., protocols or topologies. The possibility to investigate different variants in a fast and flexible way gives the opportunity to perform optimizations and find the best or improved variants that use fewer resources with respect to the number of devices, necessary device quality, memory or time.

[0191] Improved readout using global operations and multiple copies. Readout and measurements in the quantum simulation are not restricted to be local operations, as they would be in a real-world quantum network. This makes the analysis of the resulting states significantly more efficient, since more complex measurements can be realized in the quantum simulation with a small overhead, whereas this is extremely challenging if not impossible in real networks. In a quantum computer, all qubits are close to each other, and one can perform an entangling unitary operation followed by a projective measurement on individual qubits - which corresponds to a general measurement in an entangled basis.

[0192] In principle, also multiple copies of a quantum network can be simulated on a large enough quantum computer. Having access to multiple copies also allows for more efficient and improved read-out and determining properties of the network. III. Mathematical tools III.A. Quantum channels and their representations

[0193] Any quantum operation ℰ̂ that transforms density operators into density operators iscalled a quantum channel. That is to say, ℰ̂: ℒ(ℋ^^) → ℒ(ℋ^^), where ℒ(ℋ) is the set of linear34296P-EP operators on a Hilbert space ℋ, is a quantum channel if for all density operators ^^ the operator ℰ̂(^^) is also a density operator. Note that any linear, trace-preserving and completely positive map is a quantum channel. Sometimes, in particular in communication settings, a quantum channel is understood in a narrower sense as a noise map that ideally corresponds to an identity operation. In the following, however, we do not distinguish between quantum channels in this narrow sense and general noisy quantum operations (that may ideally correspond to some arbitrary unitary operation), as both are described by a completely positive map (CPM). In particular, all methods and results we describe in the following apply to arbitrary CPMs. The terms “quantum channel” and “completely positive map” are used interchangeably herein.

[0194] There exist different ways of representing a quantum channel:

[0195] First, in the Kraus representation a quantum channel is described by the so-called Krausoperators, i.e., ℰ̂ = {^^^^}, which fulfilThe action of a quantum channel on an arbitrary state ^^ is given by ℰ̂(^^) = ∑  ^^^^^^^^^†^ . ^^ Note the Kraus representation is not unique, as given a unitary matrix U with elements (^^)^^^^= ^^^^^^, the representation with Kraus operators= ∑^^ ^^^^^^^^^^}^^is an equivalent characterization of the same quantum channel ℰ.̂

[0196] Second, the Stinespring representation, which follows from the Stinespring (or dilation) theorem, states that any quantum channel can be understood as a unitary operation Λ acting ona larger Hilbert space. A channel→ ℒ(ℋ^^) can be conceived as a unitaryΛ: ℒ → ℒ(ℋ^^ ⊗ ℋ^‾^) that also maps the environment (^‾^) evolution, where part ofthe quantum state information is potentially leaked due system-environment interactions. The action of ℰ̂ on an arbitrary state can be described by tracing out ℋ^‾^, i.e.,34296P-EPThe channel in the Stinespring representation is given by ℰ̂ = (Λ, |0^^0|^‾^). In Appendix A weshow how the Stinespring and Kraus representations uniquely relate to each other.

[0197] Third, the Choi representation of a quantum channel ℰ̂^is defined by the Choi state of the channel, denoted as Φℰ̂. It is expressed as:where |Φ^ represents a bipartite maximally entangled state given byTheChoi state offers a comprehensive description of the channel, enabling direct extraction of the Kraus operators. Asis a density operator, it can always be decomposed as:where ^^^^forms a probability distribution and {|^^^^^} represents a set of normalized states. Byexpressing= ^^^, ^^ ∣ ^^^^^, it is evident that ^^^^ =constitutes a Kraus representation of ℰ̂. Notably, this decomposition aligns with the non- uniqueness of the Kraus representation, as observed in Eq. (1). We denote the Kraus rank of ℰ̂ as ^^, which is determined by the rank of Φℰ̂, corresponding to the minimum number of elements in a Kraus representation of the channel ℰ̂.

[0198] Fourth, the Liouville superoperator representation of a quantum channel ℰ̂ encapsulates the complete description of the channel in a matrix known as the Liouville superoperator, defined as (note we describe these channel matrices with no-hat notation):This representation is particularly significant as it facilitates channel concatenation through matrix multiplication. Specifically, the Liouville superoperator of the concatenation of multiple channels corresponds to the matrix product of their respective Liouville superoperators, i.e., ifℰ3̂ = ℰ2̂ ∘ ℰ1̂, then ℰ3 = ℰ2 ⋅ ℰ1. Moreover, the action of a channel on an arbitrary state ^^ isgiven by ℰ|^^^^, where |^^^^represents the vectorized form of ^^.34296P-EP

[0199] For a given quantum channel, its Choi matrix and its Liouville superoperator are connected through a reshuffling of their matrix elements, enabling easy conversion between the two representations.

[0200] The distance between two channels, ℰ1̂ and ℰ2̂, can be evaluated using various metrics. In this work, we utilize the Choi fidelity, i.e.,The channel infidelity between two channels, ℰ1̂ and ℰ2̂ is defined as 1 − ^^(ℰ1, ℰ2).III.B. Quantum hardware noise

[0201] Although the methods of the present disclosure can be used for completely general quantum network simulation, in the following particular examples of noise models and noise types are considered, based on realistic sources of noise observed in current quantum hardware. Irrespective thereof, it shall be understood that the methods described herein are not restricted to these exemplary noise models, which are provided for illustrative purposes. III.B.1. Noise models

[0202] Two exemplary methodologies for modeling hardware noise within quantum computers (e.g., source noise as described herein) during a simulation of a quantum network protocol, namely a gate noise model and a block noise model, are described in the following.

[0203] The gate noise model involves representing the noise acting on each elementary gate performed by the quantum computer. Specifically, each noisy gate may be modelled by first implementing the ideal version of the gate, followed by the introduction of noise channels affecting each register (i.e. each quantum constituent) involved. Typical noise channels considered include dephasing, depolarizing, or amplitude damping noise, which are prevalent in current quantum devices.

[0204] Fig. 15(a) shows an illustration of the gate noise model. A quantum circuit including gates U1to Uk, e.g. unitary gates, may be performed by the quantum computer. The gates U1to Ukas such are considered to be ideal gates, i.e. free of noise. At the end of the quantum circuit,34296P-EP a measurement may be performed. The sequence of gates U1 to Uk, followed by the measurement, may realize a quantum channel ^̂^. The quantum channel ^̂^ may constitute a quantum simulation of another quantum channel (e.g. an effective completely positive map as described herein) that is part of the quantum network protocol under consideration. Particularly, the sequence of gates U1to Uk, followed by the measurement, may represent an ideal implementation of the quantum channel ^̂^, i.e. in the absence of noise. To model a noisy version of the quantum channel ^̂^, in the gate noise model it is considered that each of the gates Ui is directly followed by a respective noise channelas shown in Fig. 15(a). The gate Ui followed by the noise channel ℰimodels a noisy version of the gate Ui. The quantum circuit obtained in this manner, namely the sequence of gates U1 to Uk interspersed with the respective noise channelsto ℰkas shown in Fig.15(a), models a noisy implementation of the quantum channel ^̂^. In an illustrative example, the quantum channel ^̂^ may be the amplitude dampingwhere Y is a Pauli ^y operator acting on the second qubit, and where CNOT21represents a controlled-NOT gate where the second qubit is the control qubit and the first qubit is the target qubit. The circuit shown in Fig. 15(a) then illustrates a representation of a noisy amplitude damping channel.

[0205] The block noise model involves modelling a total, effective noise acting on an entire quantum circuit, rather than modelling each noisy quantum gate in the circuit individually. That is to say, in the block noise model, an ideal version of the entire quantum circuit (i.e. a block of ideal gates) is followed by a single noise map. Practically, this model can prove highly beneficial when knowledge of the noise acting on the individual quantum gates is unavailable. In such cases, one can model the noise effect as the ideal map implementation followed by an encompassing noise channel, effectively accounting for all noise sources during computation.

[0206] Fig.15(b) shows an illustration of the block noise model. The model involves the same sequence of gates U1 to Uk that (when followed by the measurement) realize the quantum channel ^̂^. In the block noise model, a noisy version of ^̂^ is modelled by an ideal implementation of the entire channel ^̂^, i.e. an ideal implementation of the entire sequence of gates U1to Uk, directly followed by a single noise map ℰ1, as illustrated in Fig.15(b).34296P-EP

[0207] Whereas the discussion above describes noise models for noise occurring in a quantum computer (source noise as described herein), the models in question, i.e. the gate noise model and block noise model, may likewise be used to model noise occurring within a quantum network (target noise as described herein). III.B.2. Examples of noise types

[0208] For the sake of concreteness, three illustrative examples of noise channels are described in the following. These examples may be instances of noise occurring in a quantum computer (source noise), such as the noise channels ℰidescribed with respect to Figs.15(a)-(b), but also represent possible examples of noise that may be present within a quantum network (target noise).

[0209] Dephasing noise. This type of noise is a significant factor contributing to the loss of coherence in quantum systems. It manifests itself in various quantum scenarios, such as imperfections in optical fibers or fluctuations in electromagnetic fields. Mathematically, it is defined as follows:where ^^^^denotes the ^^-Pauli operator and where p is a parameter in the interval [0, 1]. The parameter p is an example of a noise parameter as described herein.

[0210] Depolarizing noise. Also referred to as white noise, depolarizing noise holds particular relevance as it often enables the analysis or estimation of worst-case scenarios. Its expression can be represented as:The parameter p is an example of a noise parameter as described herein. Alternatively, ℰ(̂^^)can be equivalently expressed as: ℰ̂(^^) = ^^′^^ + (1 ′where ^^ = (4^^ − 1) / 3 and I isthe identity operator. This transformation maps the state ^^ to a completely mixed state withprobability (1 − ^^′), implying the loss of all state information with that probability. Notably,the depolarizing channel represents a comprehensive scenario for single-qubit channels, as any34296P-EP other channel or state can be transformed into a depolarizing form through random operations, a process known as depolarization.

[0211] Amplitude damping noise. This noise type is a primary example of a non-unital noisy channel, providing a representation of important phenomena such as spontaneous emission. The Kraus operations defining this noise are as follows:The parameter ^^ is an example of a noise parameter as described herein. It is repeated that, while the above illustrative examples are considered for the sake of concreteness, the methods described herein are general and apply to arbitrary types of noise. For example, the methods of the present disclosure can be seamlessly integrated into a broader variational framework (see Sec. V.C below) to ensure independence from specific noise forms and models. IV. Tools for the simulation of channels and measurements

[0212] For achieving a reliable simulation of quantum network protocols, it is beneficial that quantum channels and (noisy) measurements can be simulated with high fidelity. The methodologies discussed in the following encompass tools for simulating noise in a quantum network.

[0213] In this section, the case of an ideal quantum computer is considered, where unitary quantum gates, projective measurements, and state preparation processes are free of noise. Despite the idealistic nature of this assumption, its analysis serves as a foundational framework for understanding the implementation and manipulation of realistic quantum channels and measurements - described by completely positive maps (CPMs) or positive operator-valued measures (POVMs), respectively - within a quantum circuit architecture. Strategies for managing and leveraging source noise present in the quantum computer are subsequently addressed in Sec. V.

[0214] We analyze how an arbitrary ^^-qubit quantum channel can be implemented (simulated) within a perfect quantum computer. We consider a quantum computing system capable of executing unitary gates, projective measurements, and stochastic mixtures of both, without noise. Any channel will be implementable by the quantum computer as long as the quantum34296P-EP computer operates on a quantum system (a set of quantum constituents as described herein) possessing a sufficiently large Hilbert space (e.g. the number of qubits shall be sufficiently large). Within this section, we examine the spatial overhead ^^^^necessary to implement a general ^^-qubit channel, defined as the number of ancillary qubits required by the quantumcomputer, i.e., ^^^^ = log2^^^ − ^^, where ^^ represents the dimension of the Hilbert space of thequantum computer. Given that the state of an ^^ qubit system is equivalent to that of a quditsystem with dimension ^^ = 2^^, we restrict our focus in this section to single-qudit channels.

[0215] We explore two distinct approaches: firstly, an ancillary-assisted quantum computer configuration, where the quantum computer comprises ^^ qubits (being the quantumconstituents in the present example), thereby setting ^^ = 2^^; and secondly, a scenario wherethe processor can make use of an arbitrarily large qudit system, denoted as ^^ ∈ ℕ.IV.A. First approach: ancilla-assisted

[0216] In the first approach, which we denote as the ancilla-assisted approach, simulating quantum channels is achieved directly through unitary gates and projective measurements. According to the Stinespring dilation theorem (refer to Sec. III.A.1), any quantum channel or completely positive map (CPM) ℰ̂ can be represented as a unitary transformation Λ acting on an extended system. By simulating this unitary transformation and subsequently tracing out the additional ancillary system(s), one can effectively implement any quantum channel by aquantum computer. In this approach, the spatial overhead is determined by ^^^^ = log2(^ ^^^^),where ^^^^denotes the dimension of the ancillary system(s).

[0217] In Appendix A, an explicit procedure is provided for determining the unitary gate Λcorresponding to any given quantum channel. The dimension of Λ is given by dim^(Λ) = ^^^^,where ^^ represents the Kraus rank of the channel and where d is the Hilbert space dimension,in this case ^^ = 2^^ with n the number of qubits on which the channel acts. As the Kraus rankof a qudit channel is upper-bounded by ^^2, this method enables the implementation of any ^^-qubit channel with an overhead of ^^^^ = 2^^. Consequently, the quantum computer shall becapable of manipulating ^^ = 3^^ qubits in total, i.e. the n qubits in question plus 2n ancillaryqubits (also called auxiliary qubits). This is illustrated in Fig.16(a) (see the middle part of the figure).34296P-EP

[0218] Further, the minimum ancilla requirements for stimulating a CPM using extreme quantum channels are considered. The set of quantum channels is convex, meaning that any arbitrary channel ℰ̂ can be expressed as a convex combination of extreme channels. Mathematically, this can be represented as:where 0 < ^^^^ ≤ 1,∑^^  ^^^^ = 1. Here, ℰ^̂e^xtdenote the extreme channels, which are those that cannot be expressed as convex combinations of other quantum channels. Therefore, one can infer from Eq. (5) that ℰ̂ can be realized by implementing ℰ^̂e^xtwith probability ^^^^.

[0219] The Kraus rank of any extreme channel is at most ^^, where ^^ represents the dimensionof the Hilbert space (where, again, ^^ = 2^^ with n the number of qubits on which the channelacts). Therefore, an ancillary system of dimension ^^ is sufficient to implement any ℰ̂extand,thus, any quantum channel ℰ̂. This insight leads to an overhead of ^^^^ = ^^, meaning that thetotal number of qubits can be reduced to ^^ = 2^^ qubits, i.e. the n original qubits plus n ancillary(auxiliary) qubits. This is illustrated in Fig.16(a) (see the right-hand part of the figure).

[0220] It may be beneficial to further minimize the resource requirements for channel simulation. In this regard, employing techniques such as stochastic maps or classical mixtures of unitary operations can facilitate the implementation or approximation of various types of quantum channels even without the need for ancillary systems. In the subsequent discussion, we show relevant cases where utilizing stochastic mixtures of unitary gates enables a reduction in the size of the quantum computer. IV.A.1. Examples

[0221] Here we elaborate on the above-discussed distinctive features for various particular types of completely positive maps (CPMs), where the need for ancillary systems required to simulate them can be minimized or even disregarded.

[0222] Unital channels. A notable category is that of mixed-unital channels, denoted as ^̂^, which consist of convex combinations of unitary gates. Mathematically, they are expressed as:34296P-EPwhere {^^^^} form a probability distribution and each ^^^^represents a unitary gate. These channels can be implemented by applying gate ^^^^with probability ^^^^, without the necessity for ancillarysystems. Noteworthy noisy channels falling into this category include bit flip noise (^̂^ ={^^^^, (1 − ^^)^^}), phase damping (^̂^ = {√^^^^^^^^^^^^^^} ^^ ), and depolarizing noise, discussed in Sec. III.A.

[0223] Projective measurement channels. Another class of channels that can be realized without ancillary systems are those that can be decomposed into a projective measurement followed by a correction operation. These channels possess a Kraus representation of the formℳ̂ = {^^^^^^^^}, where ^^2^^ = ^^^^ denotes a projector. Implementing channel ℳ̂ involves performinga measurement described by{^^^^}followed by the correction operation ^^^^, potentially discarding the measurement outcome. An important example of this type is the quantum reset channel (ℛ̂), which is a specific instance of an extreme channel that can be implemented without requiring ancillary.

[0224] Erasure channel. The quantum erasure channel models scenarios where the physical particle encoding quantum data is lost with a certain probability ^^, such as photon loss in optical systems. This channel transforms a qubit state ^^ as follows:where |^^^ is an orthogonal state to ^^. The erasure channel can be simulated with a single ancil- lary system in the pure state |0^^^, by leaving the state untouched with probability ^^, and pro-jecting the system to the state |0^|1^^^ with probability (1 − ^^), i.e.,In this scenario, the ancillary states are not traced out but are retained throughout the computa- tion.34296P-EP IV.B. Second approach: extended qudit

[0225] Here the benefits derived from employing higher-dimensional systems (^^-level systems or “qudits”) are described, referred to herein as the extended qudit approach, where different quantum levels of the qudits can play the role of ancillary registers. We investigate the resource requirements for simulating arbitrary quantum channels in this manner, noting that using this strategy with various dimensions of qudits can be particularly advantageous for simulating certain channels.

[0226] Although in the extended qudit approach quantum information is still encoded in qubits, these qubits are embedded within a subspace of the ^^-level system, which we denote as the data subspace. Consequently, the quantum computer can utilize the additional Hilbert space (i.e. the remaining quantum levels of the qudit) to dilate any quantum channel into a routine comprising unitary operations and projective measurements. The extended qudit approach can lead to more efficient channel implementations, as only single-qudit operations are required instead of e.g. two-qubit interactions (qubit-ancilla). Importantly, experimental research has demonstrated that individual qudits can be fully manipulated with comparable accuracy to a single qubit. Additionally, as detailed below, employing qudit systems can reduce the simulation overheads compared to the ancilla-assisted approach.

[0227] The extended qudit approach is illustrated in Fig.16(b).

[0228] It is demonstrated how, given an arbitrary quantum channel, one can derive a routine ^̂^ that implements it in a qudit quantum computer. For mathematical convenience, we assume thatquantum information is encoded in a qudit state ^^ ∈. This qudit state is implementedwithin an extended qudit system of a larger dimension D, where the state of the entire systemis represented by ^^ ⊕ ^^ ∈ ℒ(ℂ^^).

[0229] Given a qudit channelwe initially apply a unitary gate Λ that encodesin different orthogonal subspaces of the Hilbert space of the extended qudit system, where ^^^^is a unitary gate. Subsequently, we perform a projective measurement{^^^^}^^^^=−01into each of these subspaces. This measurement leaves the state of the 1 entire system in the state ^^^^^^^^†^^^^^^^^^†^ in the corresponding subspace with probability ^^^^=tr[^^^^^Λ(^^ ⊕ ^^)Λ†]. We then implement a correction operation ^‾^ ^^, which first returns the state34296P-EP of the qudit back into the data subspace and then inverts. At this stage, the state of the system is given bywhere refers to the branch of routine ^̂^ where outcome ^^^^is obtained. Finally, the measure- ment outcome is erased, leading to a convex mixture of all branches, i.e.,Refer to Appendix B for detailed explanations.

[0230] The dimension of the subspace required to encode ^^^^†^^†^^^^^^^^ ^^^^ is given by rank(^ ^^^^) =^^^^. Therefore, the quantum channel can be simulated if ^^ ≥ ∑^^^^=−01 ^^^^. It is noteworthy that ∑^^−1^^=0  ^^^^ indicating that the utilization of qudits can result in a reduced spatial overheadcompared to ancillary systems.

[0231] Example. Consider the amplitude damping channel ^̂^ from Eq. (4). In this case, thethe ranks of the Kraus operators is 3, indicating that it suffices to implement thequbit in a 3-level system. Thus, the input state is given by dim^(^^ ⊕ 0) = 3 × 3. First, we applythe unitary gatefollowed by the projective measurement with ^^0 = |0^^0| +and ^^1 = |2^^2|, along withthe corresponding correction operations ^‾^0 = ^^ and ^‾^1 == |(^^ + 1)mod^^^.As a result, we obtain the state ^̂^(^^) ⊕ 0. Therefore, the overhead is given by ^^1 = 1 −log2^3 ≈ 0.58.

[0232] On the contrary, if we utilize the ancilla-assisted approach, an ancilla qubit is required,resulting in an overhead of= 1. This comparison illustrates an example of the advantage34296P-EP employing the extended qudit approach, as it leads to a reduction in overhead compared to the ancillary-assisted approach. IV.C. POVM simulation

[0233] The approaches outlined above for simulating quantum channels can also be extended to simulate noisy measurements, as described by a Positive Operator-Valued Measure (POVM).Given a POVM ^^ =it can always be implemented as a von Neumann measurement acting on a larger Hilbert space. If the input state is considered to be given by^^ ⊕ ^^, then ^^ can be performed by implementing channel ℰ̂ =as described previously. However, in this case, the outcome of the projective measurement on the qudit system is not erased but learned. Therefore, the dimension of the extended qudit system mustsatisfy ^^ ≥ ∑^^^^=0 rank^(^^^^). V. Simulation of quantum network protocols with noisy quantum computers

[0234] In any realistic quantum computing hardware, NISQ devices in particular, noise and decoherence are inevitable factors that affect the reliability of quantum computations.

[0235] One possible approach for realizing a quantum channel using a noisy quantum computer is to directly utilize the tools described in Sec. IV. When the quantum computer is noisy, the implementation of a quantum circuit for realizing a desired quantum channel will no longer be perfect. Hence, the desired quantum channel will not be implemented perfectly either. Rather, an approximation of the channel will be realized.

[0236] However, in the following it is demonstrated how such a direct approach can be significantly improved by incorporating the noisy characteristics of the quantum computer into the simulation. For achieving a faithful simulation of a quantum network protocol on quantum hardware involves, diverse tools and techniques are described. These tools enable the replication of quantum schemes within a quantum circuit while effectively managing - or even leveraging - the inherent source noise present in the quantum computer.

[0237] As already described before, a general task is to obtain a quantum simulation of a quantum channel that forms part of a quantum network protocol (e.g. an effective completely positive map as described herein). We denote by ℰt̂ the desired channel to be implemented on34296P-EP the quantum hardware. That is to say, ℰt̂ is a channel that, when performed perfectly by the quantum computer, i.e. in the absence of noise, provides a quantum simulation of a particular quantum channel that is part of the quantum network protocol. While an ideal, noiseless quantum computer could implement the channel ℰt̂^with perfect fidelity using the techniques discussed in Sec. IV, imperfect quantum hardware necessitates the development of strategies to account for source noise, exploit them, or transform certain quantum channels.

[0238] Generally, we do not aim for reducing source noise. Our approach typically involves increasing the amount of noise in gates and processes, since quantum channels that connect quantum devices are typically noisier than local systems, and local nodes in a network are either themselves small-scale quantum computers, or systems with increased functionality (e.g., also include an interface to photonic systems to allow for communication) and hence have a lower quality than a specialized quantum computing device. Furthermore, we may directly utilize the existing source noise and reshape its form to match one of the desired target noise processes. This is in fact a simpler and remarkably less demanding process than noise mitigation or trusting in protocols that are based on noiseless devices, and can be more easily performed using existing NISQ devices. Still, the accuracy can profit from advances in the control and quality of quantum computers.

[0239] In the following, several general methods are provided to address these challenges and we evaluate their effectiveness through relevant examples.

[0240] Firstly, the building-block method is described, where circuits simulating the desired channels ℰt̂^are conceived as fixed building blocks. In this approach, the inherent structure of the simulation circuit - such as the sequence of gates - is fixed and inaccessible for direct modification. However, the circuit is affected by (source) noise, resulting in imperfect implementation. To counteract this, we introduce supplementary channels, also implemented within a quantum circuit framework and also subject to (source) noise. These additional channels can be strategically applied before, after, or via classical mixture to compensate for the noise effects and approximate the overall circuit to the desired implementation.

[0241] Secondly, we explore the tailored circuits method, where circuits intended for channel simulation are not fixed. Here, one can adjust the circuits to approximate the desired channel as closely as possible, considering hardware imperfections. We demonstrate instances where34296P-EP the direct implementation of channel simulation circuits, which is optimal in noiseless scenarios, becomes suboptimal when noise is accounted for. Additionally, we present various scenarios with differing flexibility ranges for optimizing the quantum circuit.

[0242] Lastly, we propose an extension that can complement or integrate with the aforementioned strategies (and tools from Sec. IV), leveraging variational quantum optimization techniques. These techniques involve quantum algorithms that enable the simulation of a quantum channel with only accessible input parameters, without requiring detailed knowledge of the simulation's internal workings. By maximizing the fidelity of the simulated process through appropriate optimization of these input parameters, we achieve an approximation of the desired simulation directly. We refer to this strategy as the Variational black-box method. It is essential to address preparation and measurement errors and design efficient procedures for measuring channel fidelity.

[0243] Real devices can be seamlessly integrated into simulations either via direct interface replacement or through benchmarking to obtain an optimized description within a specified error model. V.A. Method 1: Building-block channels

[0244] The first strategy we propose is denoted as the building-block method, illustrated in Fig. 17(a). In this method, a quantum circuit used to realize (an ideal version of) a particular quantum channel ℰt̂^^is treated as a fixed building block. The sequence of gates making up the quantum circuit is as such not directly modifiable. Each of the gates in the quantum circuit will be subject to noise (source noise), so that the quantum computer will perform a noisy version of the quantum circuit or, in other words, a noisy version of the quantum channel ℰt̂^. Whereas the structure of the quantum circuit is treated as being fixed, it is however possible to introduce supplementary channels. These supplementary channels, which are also susceptible to (source) noise, are strategically applied in various configurations. For instance, they can be applied before the quantum circuit, after it, or both (referred to as interleaved). Additionally, the method allows for the consideration of nontrivial classical mixtures of these supplementary channels applied with different probabilities. The addition of the supplementary channels is performed in a manner so that the total quantum process, i.e. the noisy quantum circuit plus the supplementary channels, achieve a satisfactory approximation of the desired channel ℰt̂.34296P-EP

[0245] Concretely, as illustrated in Fig. 17(a), the building-block method implements a quantum channel ℰôutof the following form:Therein, {^^^^^^} defines a probability distribution, ℰt̂′is a noisy version of the quantum channel ℰt̂ (i.e. ℰt̂′is the channel that is actually performed by the quantum computer, since the quantumcircuit realizing ℰt̂^is subject to source noise),are supplementary quantum channels thatcan be freely chosen, and the prime notation, i.e.indicates the realistic, i.e. noisy implemen- tation of the channel under consideration. To represent the noise, any of the models used in Sec. IIIB can be used.

[0246] Within this setting, an optimization (typically a classical optimization) is performedover the possible supplementary channelsso that the resulting channel ℰôut maximizesthe channel fidelity ^^(ℰôut , ℰ^̂^) between ℰôut and ℰ^̂^ such that ℰôut ≈ ℰt̂ , where, again, ℰ^̂^ isthe ideal quantum channel that one actually wishes to implement by the noisy quantum computer.

[0247] Utilizing the superoperator representation, as discussed in Sec. III.A.1, channel concatenation can be expressed as matrix multiplication. With this formalism, the optimizationproblem reduces to finding matricesthat maximize the channel fidelity while consideringnoise (i.e., accounting for their imperfect implementation ^^′^^, ^^^′^). These matrices must represent valid physical quantum channels, meaning that the corresponding channels (or Choi matrices,see Sec. III.A.1)adhere to positivity, trace preservation, and complete positivityproperties. An optimal set of channelscan be found, for example, by a numericalanalysis. Various optimization techniques can be used, such as interior point optimization or active set optimization, or any other optimization techniques (and the same techniques can be applied to other instances of optimization problems described elsewhere herein). While the examples presented herein focus on specific scenarios, the techniques are broadly applicable.

[0248] Fig.18(a) presents an illustrative example where ℰt̂ is a bitflip channel with 5% noise strength, which is realized using its basic circuit implementation. In other words, the parameter p representing the probability of a bit flip (represented by the Pauli operator X) is set to a fixed34296P-EP value of p = 95%; for further details, see appendix C. In this example, the bit flip channel is subject to noise ℬ̂. The noise ℬ̂^is modeled using the block noise model discussed in Sec. III.B,such that the circuit is ideally implemented, followed by the noise ℬˆ. Specifically, in thisexample the noise ℬ̂^^^^^ characterized by Kraus operatorswiththe noise parameter ^^ characterizing the strength of the noise. The choice of the noise ℬ̂^is motivated given its non-Pauli diagonal nature, in order to illustrate how our method works in these scenarios too.

[0249] The objective here is to compensate for the introduced noise ℬ̂^by employing the interleaved building-block strategy. Each supplementary channel introduced to compensate for the noise is subjected to the same amount of noise ℬ̂, modeled likewise according to the block noise model. The challenge lies in designing these additional channels in such a way that they effectively mitigate the noise ℬ̂^present in the bit flip channel. For comparative and fundamental purposes, we also analyzed the ideal scenario where the additional channels are not subjected to any noise.

[0250] Fig. 18(a) shows the results of applying the above-described building-block approach to the present example. The x-axis 1800 represents different values of the parameter q, i.e. the strength of the noise ℬ̂. The y-axis 1801 represents the optimal (= minimal) channel infidelity (being equal to 1 – F, where F is the channel fidelity) between the desired channel ℰt̂, i.e. the bit flip channel in the present example, and the channel ℰôut, optimized over the supplementarychannels {^^^^ , ^^^^}. Plot 1802 is a reference plot that shows the channel infidelity in the case wherethe supplementary channels are absent, in oth′er words in this case we have ℰôut = ℰt̂. Plot 1804 shows the optimal channel infidelity in the case where the supplementary channels are performed perfectly, i.e. free of noise. Plot 1806 shows the optimal channel infidelityin the realistic case where the supplementary channelsare noisy, i.e. subject to the noiseℬ̂, like the bit flip channel itself. As can be seen, for a substantial range of values for the parameter q, plot 1806 lies below plot 1802, so that the channel infidelity can be reduced byperforming suitable supplementary channelseven if these channels are noisythemselves.34296P-EP

[0251] The present example highlights the potential advantages of employing the building- block approach in specific regimes. Even when each supplementary channel is also subjected to the noise ℬ̂, the interleaved strategy (plot 1806) demonstrates its effectiveness in mitigating noise-induced errors. This observation underscores the potential utility of the approach in practical quantum simulation using NISQ devices scenarios. Further technical details of bit-flip channel implementations are provided in Appendix C.

[0252] Instead of solely compensating for noise, this method can also be utilized for channel transformation (“re-shaping”), with the objective of converting an input channel ℰîninto a potentially very different target channel ℰt̂ using the aforementioned techniques. In the context of simulating a quantum network protocol, there may arise scenarios where a specific type of noise occurring in the quantum network needs to be simulated from a particular point. The above-described building-block strategy enables the transformation of the (source) noise present in the quantum computer into the desired form, effectively harnessing the inherent source noise. From a fundamental perspective, the task of transforming one type of noise into another using additional channels applied before and after holds significant importance, even within the context of noiseless quantum hardware assumptions.

[0253] The problem of transforming channels into other channels is closely related to the concept of channel decomposition. Channel decomposition analyzes how a given channel can be expressed as a composition of other channels. It has been demonstrated that any quantum channel can be decomposed into an arbitrary number of other channels. In this context, our aim is to address this problem by employing a minimal number of additional channels, typically just two channels acting before, after, or both before and after the input channel, in the manner described above. This approach allows us to achieve channel transformation with minimal resources. For example, transforming a bit-flip channel into a dephasing channel and vice versa can be achieved by simply applying a Hadamard unital channel before and after the input channel (up to inherent noise). However, we also consider scenarios involving highly non- trivial transformations, particularly those between unital and non-unital channels.

[0254] Figure 18(b) illustrates the effectiveness of this approach in transforming, or re-shaping, an amplitude-damping channel (being the input channel ℰînin the present case) into a depolarizing channel (being the target channel ℰt̂ in the present example). Specifically, anamplitude damping channel with a fixed ^^ = 0.1 is considered (see Eq. (4)), and the parameter34296P-EP p in the depolarizing channel is allowed to vary arbitrarily. Figure 18(b) presents results obtained under both noiseless and noisy quantum computer operations. In the noisy case, the source noise is now only depolarizing noise with fixed parameter q = 0.9. The noise is again modeled using the block noise model described in Sec. III.B.

[0255] In Fig. 18(b), the x-axis 1810 represents the values of the parameter p for the target channel ℰt̂. The y-axis 1811 represents the channel infidelity, analogous to Fig.18(a). Plot 1812 is a reference plot that shows the channel infidelity in the case where the supplementarychannels are absent. Plot 1814 shows the optimal channel infidelity in the case wherethe supplementary channelsare performed perfectly, i.e. free of noise. Plot 1816 showsthe optimal channel infidelity in the realistic case where the supplementary channelsarenoisy, i.e. subject to the depolarizing noise. As can be seen, for a substantial range of values for the parameter p, plot 1816 lies below plot 1812, so that the channel infidelity can be reducedby performing suitable supplementary channels {^^^^, ^^^^}, even if these channels are noisythemselves.

[0256] Thus, remarkably, the ability to apply block channels in an interleaved fashion demonstrates significant tunability of the different channels across most regimes. This holds true even when the additional channels are subjected to noise, highlighting the robustness and versatility of the approach.

[0257] Further extensions, for instance, considering a larger number of concatenated channels (before and after), can be considered to further enhance the accuracy of these strategies. V.B. Method 2: tailored circuit channels

[0258] In the realm of simulating quantum channels by quantum circuits, we describe how manipulating gate sequences or adjusting gate parameters can lead to more accurate channel implementations, either to harness and convert or directly compensate for the source noise in the quantum computer. In this method, we propose a flexible strategy (called “tailored circuit channels”) where the circuits realizing the desired channels are not rigidly defined. Instead, one has the freedom to shape these circuits to effectively incorporate the hardware noise and better approximate the desired channel implementations.34296P-EP

[0259] An instance of the tailored circuit channels approach is illustrated in Fig. 17(b). The present approach considers a quantum circuit where one or more quantum gates Ui in the quantum circuit depend on a parameter ^i. In a realistic scenario, the quantum circuit will be noisy, i.e. the gates are subject to source noise (which may be modeled in any manner as described herein). Then, for such a noisy circuit, an optimization algorithm (typically a classical optimization algorithm) is performed over the parameters ^1, ^2, ... in order to look for an optimal configuration of parameters such that the noisy circuit provides a good approximation of a desired quantum channel ℰ̂.

[0260] In one example, the optimization may start from an initial configuration of the parameters ^1, ^2, ... such that the corresponding quantum circuit, when realized perfectly, i.e. without noise, provides an exact implementation of the desired quantum channel ℰ̂. Then, a noisy version of the quantum circuit may be considered, and the parameters ^1, ^2, ... may be varied, starting out from the aforementioned initial configuration, in search of a parameter configuration that, in this noisy setting, provides an optimal approximation of the desired quantum channel ℰ.̂ As will be demonstrated below in several examples, the inventors have found that in many cases that the optimal configuration of the parameters deviates from the initial configuration. Thus, the optimization performed over these parameters is beneficial since it can improve the accuracy for realizing the desired quantum channel ℰ̂ when the quantum computer is affected by source noise.

[0261] Through both relevant analytical and numerical examples, we illustrate the efficacy of this approach. Analytically, we explore various possibilities considering different hardware noise assumptions, where we demonstrate how varying specific parameters within the quantum gates that implement these channels, it is possible to compensate for source noise and significantly improve simulation accuracy. Numerically, we further investigate this method by considering two distinct scenarios. In the first scenario, we allow for complete flexibility in shaping the circuits, enabling adjustments to all gate parameters. In contrast, the second scenario imposes constraints, permitting only certain parameters of the gates to be tuned.

[0262] By comparing the results of these numerical analyses, we gain valuable insights into the practical implications and benefits of this tailored circuit strategy.34296P-EP Analytical example 1. Pauli-diagonal noise

[0263] The present example starts out by applying an n-qubit Pauli diagonal channel ℰîngiven bywhere each Σ^^is an element of the ^^-Pauli group ^^^^and where the pi form a probability distri- bution. The channel ℰîn^may, for example, represent source noise acting on the quantum state ^^ during a period where no gate is applied.

[0264] Further, the desired channel ℰ̂^that is intended to be realized by a noisy quantum circuit is another Pauli diagonal channel. We show that the channel ℰ̂ can be obtained by transforming the initial channel ℰîn^by applying specific unitary operations after ℰîn. The unitary operations in question are noisy, i.e. subject to source noise, modeled here according to the gate noise model outlined in Section III.B. The source noise is also considered to be Pauli diagonal noise. Accordingly, the application of a gate U that is subject to source noise is represented by theapplication of the mapwhere the source noise ℰŝnis another Pauli diagonal channel. For the sake of illustration, the source noise ℰŝnis in the present example asingle-qubit depolarizing noise, i.e., ℰŝn = ^̂^. Likewise, the channel ℰîn is also considered tobe a single-qubit depolarizing channel.

[0265] Specifically, in order to realize^ℰ,̂ the map ℰînis followed by applying noisy Pauli gatesℰŝn(Σ^^ ∙ Σ^^) with probabilityafter the application of ℰîn. Therein, Σ^^ are Pauli gates (whichare unitary operations that are equal to their Hermitian conjugate) that are noisy, i.e. subject tosource noise ℰŝn, so that ℰŝn(Σ^^ ∙ Σ^^) is applied. The probabilities ^^^^ are considered to be tunableparameters (corresponding to the parameters ^1, ^2, ... above), that are suitably selected to obtain an optimal approximation of ℰ̂.

[0266] Mathematically, ℰînis thus transformed as34296P-EPwhere {^^^^} is another probability distribution. Since the resulting channel remains Pauli diago- nal with coefficients ^̃^^^, the process can be adapted, through appropriate selection of ^^^^, to achieve the desired channel ℰ.̂

[0267] In the particular case where the hardware Pauli diagonal noise ℰŝncorresponds to singlequbit depolarizing noise, i.e., ℰŝn = ^̂^ and likewise ℰîn = ^̂^, we observe that ^̃^^^ = ^^^^^^^^ +(1 − ^^^^) / 4, where ^^ = (4^^0 − 1) / 3 and ^^ = (4^^0 − 1) / 3. Consequently, for a target Paulidiagonal channel ℰ̂ characterized by probabilities {^̃^^^}, given the condition 1− ^^^^ 1 − ^^^^4≤ ^̃^^^ ≤ ^^^^ +4is satisfied, we can perfectly realize ℰ̂ by setting= (4^̃^^^ + ^^^^ − 1) / (4PQ).Analytical example 2. Amplitude damping noise

[0268] We delve here into interesting tunability properties of the amplitude damping channel (refer also to Appendix D). In this scenario, the amplitude damping channel serves as the desired simulation implementation, and also as a model for the inherent hardware noise.

[0269] Specifically, we observe that the composition of two amplitude damping channels with noise parameters ^^1and ^^2(corresponding to the parameter ^^ in appendix D) results in yetanother amplitude damping channel, with parameter ^̃^ = ^^1 + ^^2 − ^^1^^2. Consequently, byapplying an amplitude damping channel with noise parameter P multiple times, one can simulate amplitude damping channels with higher ^̃^. It is worth noting, however, that onlydiscrete values are attainable (more if ^^ is small), and ^̃^ → 1 for multiple applications. For ^^applications, the effective noise parameter ^̃^^^can be described by a specific polynomial of degree ^^. This already suffices to obtain extremely good approximations of the desired noise process, with a fidelity ^^1.34296P-EP

[0270] A situation that cannot be easily handled with this approach involves cases where the achievable discrete values are far from the desired one, which can e.g. occur if the desired noise parameter ^̃^ is only slightly larger than the initial noise parameter ^^. In these cases, an implementation as indicated in Fig.19(a) can be beneficial. Fig.19(a) shows a quantum circuit which, when the gates are implemented perfectly (i.e. without source noise), provides an exact realization of the amplitude damping channel. The quantum circuit includes a controlled-^^^^(^^) gate into an ancilla that is initially prepared in |0>, followed by a CNOT gate and a tracing out of the ancilla. The controlled-^^^^(^^) gate depends on a parameter ^^. Choosing ^0 such thatsin^(^0 / 2) = √^̃^ gives an exact realization of an amplitude damping channel with noiseparameter ^̃^^if the quantum circuit is implemented perfectly. In a realistic scenario, the gates in the quantum circuit are subject to source noise (which may be modeled e.g. using the gate noise model). The source noise may be described by amplitude damping with parameter ^^ acting on both qubits. Then, an optimization may be performed to find a parameter ^^ which, in this noisy setting, results in the optimal approximation of the desired amplitude channel with parameter ^̃^. Notably, the optimal value for ^^ in the presence of source noise will be different from the value ^0 corresponding to an amplitude damping channel with noise parameter ^̃^ in the noiseless case. In fact, using this value ^0when the controlled-^^^^(^^) rotation is itself noisy is no longer an amplitude damping channel. Further, the channel fidelity ^^2is in this case low. One can optimize ^^ though, and obtain a fidelity ^^3. In most cases (small ^^, and ^̃^ much larger- about an order of magnitude or more) we have ^^1 > ^^3 > ^^2. However, e.g., for ^̃^ = 0.45,^^ = 0.4 one obtains ^^3 > ^^1 > ^^2.Numerical analysis

[0271] Figure 20 illustrates the advantages of employing the tailored circuit strategy for simulating different channels in a quantum device across various noise scenarios.

[0272] In Figure 20(a), we observe the performance of the above-described approach for implementing an amplitude damping channel, as described by Eq. (4), with varying strength ^^. The x-axis 2002 represents the parameter ^^ of the desired amplitude damping channel. The y- axis 2004 represents the optimal channel infidelity between the desired amplitude damping channel and the noisy quantum circuit, where the parameter ^^ is optimized over. The noisy circuit is shown in Fig.19(b), where source noise is modeled in a noisy-gate fashion, following34296P-EP the approach outlined in Sec. III.B. Specifically, noisy channels act in each register after every gate, as also illustrated in Fig.19(c) (in Fig.19(c) we assume the same noise but the controlled- X gate is substituted by a measurement followed by a conditioned X gate). Here, the hardwarenoise is fixed as depolarizing followed by dephasing, i.e. ^^^^ = ℰd̂ep^ ∘ ^ ℰd̂eph, with a strengthof 7.5% each (corresponding to p = 0.925). Adopting the tailored circuit strategy, we focus on tuning only the parameter ^^ from the gate of the circuit in Fig.19(c). Fig.20(a) demonstrates how the optimal controlled-rotation parameter ^^ varies in the presence of source noise. Plot 2006 shows the optimal channel infidelity as a function of the parameter ^^, when the parameter ^^ is optimized over. Plot 2008 is a reference plot showing the channel infidelity in case the circuit corresponding to the parameter value ^0is applied (subject to the same source noise), without performing any optimization^^^. As shown, plot 2008 lies below 2006, showing that optimizing over the parameter^^^ can reduce the channel infidelity, and thus enhance the quality of channel simulation.

[0273] In addition, Fig. 20(b) explores a similar approach but with source noise modeled according to the block noise model outlined in Sec. III.B. The source noise again consists of depolarizing followed by dephasing, but now with a strength of 20% each. The x-axis 2012 and the y-axis 2014 are defined similarly as in Fig. 20(a). Further, the plots 2016 (optimal channel infidelity when optimizing over ^^) and 2018 (reference plot) are analogous to plots 2006 and 2008, respectively, in Fig.20(a). Further, we also consider full circuit tunability (plot 2019), where each matrix element of each unitary gate can be adjusted, and the results indicate that this approach is generally even more beneficial (since plot 2019 is the lowermost plot). Notice that channel infidelities are high due to the substantial amount of inherent noise assumed in these simulations.

[0274] Finally, in Fig. 20(c), assuming again the block noise model now with dephasingfollowed by amplitude damping noise (both with noise parameter ^^ = 0.8), the objective is thesimulation of a depolarizing channel. The x-axis 2022 represents the noise parameter p of the depolarizing channel. The y-axis 2024 represents the channel infidelity, similarly as above. Similar to Fig. 20(b), we consider different tunability variants: a direct implementation (reference plot 2028), entailing the application of each Pauli operator with equal probability; Pauli diagonal optimization (plot 2026), enabling optimization over the probabilities associated with each Pauli operator; and full circuit optimization (plot 2029). Notably, even partial34296P-EP optimization, specifically tuning the probabilities of Pauli operators, yields significant enhancements compared to direct channel simulation. These enhancements can be further augmented when granting full flexibility in circuit design.

[0275] These examples underscore the potential of leveraging quantum computers for simulating quantum communication processes. Importantly, advantages are discerned across all cases, irrespective of the noise model under consideration. Furthermore, it is noteworthy that similar qualitative behavior persists across diverse hardware noise sources and analyzed regimes. V.C. Method 3: variational black-box optimization

[0276] In this section we discuss a versatile method (“Variational black-box optimization”) that complements the other proposed approaches and strategies, offering a unique perspective on circuit simulation. The core concept involves treating the realization of a desired quantum channel as a black-box optimization problem, where only certain parameters are accessible for adjustment. The optimization algorithm runs physically on a quantum computer and is hence a quantum algorithm. A quantum circuit is physically applied. As before, one or more of the quantum gates in the circuit may depend on a parameter. Further, the gates are subject to source noise, i.e. the quantum circuit is noisy. Yet, different from what was described above, in the present case the noisy quantum circuit is physically implemented as a quantum mechanical process performed by the quantum computer. The parameters of the gates may be updated iteratively in order to optimize an output figure of merit, such as the infidelity of the actually realized quantum channel as compared to the desired quantum channel. That is to say, for each configuration of the parameters of the gates, the corresponding quantum circuit is physically ran on the quantum computer, and the infidelity of the realized channel with respect to the desired channel is measured (e.g. by performing quantum tomography). This procedure is repeated for many parameter configurations. In this way, an optimal configuration of the parameters (i.e. a configuration where the channel infidelity is minimal) can be iteratively determined by suitably varying the parameters.

[0277] The variational black-box optimization is illustrated in Fig.17(c). The method may start with the preparation of a Bell state at 1701, followed by a black box channel simulation at 1702, followed by a channel fidelity estimation at 1703. The result of the channel fidelity estimation34296P-EP is used to determine a subsequent configuration of the parameters of the gates in a feedback loop as indicated at 1704.

[0278] A particularly advantageous aspect of this variational approach lies in its practicality. Firstly, it is hardware independent, eliminating the need for prior knowledge about the source noise characteristics of the quantum computer. Further, it operates in a black-box simulation fashion, where users iteratively optimize over classical parameters fed into the quantum computer, which, in turn, provides information about the figure of merit in such black-box manner. Further, this approach is fully general, and can be used to realize any arbitrary quantum channel using a noisy quantum computer, regardless of the type of source noise that might be present in the quantum computer.

[0279] To illustrate this variational strategy, consider the tailored circuit channel method (described in Sec. V.B), applied for simulating an amplitude-damping channel. The quantum circuit depicted in Fig. 19(a) can be physically performed by a quantum computer. In this physical implementation, source noise will be physically present, and hence there is no need to model the source noise classically. The noisy quantum circuit can be viewed as a black box, where the parameter ^^ of the controlled rotation ^^^^(^^) can be freely adjusted. Optimization over this parameter, based on the measured channel fidelity compared to the target channel, maximizes the quality of the simulation without necessitating knowledge of the source noise.

[0280] Similar techniques can be employed with different input parameters or optimized over arbitrary circuit implementations of channels (or network processes in general). This flexibility furnishes a potent and general simulation tool capable of operating independently of specific inherent noise sources in quantum devices. VI. Conclusions

[0281] We have presented a new strategy for utilizing quantum computers for simulating quantum network protocols, offering significant advantages over classical simulations.

[0282] Quantum computers provide an advantageous platform for simulating large networks comprised of many qubits, whose exponentially large state space renders exact classical descriptions infeasible. While classical methods rely on simplified noise models and are constrained to certain types of states, our approach is versatile, capable of simulating any34296P-EP communication process under realistic conditions, and circumventing waiting times inherent in classical simulations.

[0283] We have introduced and expanded upon various fundamental tools for simulating these networks on quantum hardware. Importantly, our approach leverages the inherent noise present in quantum computers, turning it from a limitation into an asset for our simulation purposes, rather than expending resources to mitigate or suppress it. Appendix A: Channel dilatation with an ancillary system

[0284] Stinespring dilation theorem shows that given a quantum channel ℰ̂ one can always finda unitary gatesuch thatℰ̂(^^) = tr^‾^[ ^Λ(^^^^^^|0^^0|^‾^)Λ†] (^^1)where dim(^ ℋ^‾^ ⊗ ℋ^^) = dimIn this section, we consider the case where ℋ^^ = ℋ^^ and ℋ^‾^ = ℋ^‾^ , and we show that ifdim(^ ℋ^‾^) = ^^ where ^^ is the Kraus rank of ℰ,̂ then Λ always exist.

[0285] Being {^^^^}^^^^=−01a minimal Kraus representation of ℰ,̂ if Λ fulfils ^^^, ^^|Λ|0, ^^^ = ^^^|^^^^|^^^, (^^2)or equivalently in the computational basis, it is of the formthen its action on the system is given by34296P-EPwhere ^^^^^^ = ^^^|^^|^^^.

[0286] Then, note that if the ancilla state is traced out the quantum channel is implemented to the state ^^, i.e.,

[0287] Therefore, condition Eq. (A2) guarantees that Λ is a dilation of channel ℰ̂. Now, we still need to show that the condition in Eq. (A2) is always compatible with Λ being unitary. Λ is a unitary matrix if and only if it can be written aswhere {|^^^^^^^} is an orthonormal basis. Condition in Eq. (A2) completely determines {|^^0^^^}^^^^=−01which are given by34296P-EP Note that by construction it is fulfilled thatwhere we have used that ∑^^^^=−01= ^^. Therefore, {|^^0^^^} form an orthonormal set. Then, wealways can find the rest of vectorssuch that {|^^^^^^^} forms an orthonor- mal basis, i.e., completing matrix Λ such as it is unitary. Appendix B: Channel dilatation with an extended qudit

[0288] In this section, we consider the super-qudit channel implementation, where a qudit state of dimension ^^ is encoded in the so-called data subspace given bywhere the total Hilbert space is of dimension ^^. Given a quantum channel ℰ̂, our goal is to find a routine ^̂^ consisting of unitary gates and projective measurements acting on the super-qudit that implements the quantum channel on the qudit state, i.e., ^̂^(^^ ⊕ ^^) = ℰ(̂^^) ⊕ ^^.Being{^^^^}^^^^=−01a minimal Kraus representation of channel ℰ̂, we first perform the single valuedecomposition of ^^ , and wr† ^^ ite the Kraus operators as ^^^^ = ^^^^Σ^^^^^^. Without loss of generality,we find and ^^^^ such that Σ fulfilsWith this choice, we define ^̃^^^ as aconsisting of the non zero rows of Σ † ^^^^^^, i.e.,where dim^(^̃^^^) = ^^^^ × ^^, and note they fulfil

[0289] Then, we construct a unitary matrix of the form34296P-EPwhere Λ is a unitary gate acting on a Hilbert space of dimension ^^ = ∑^^^^=−01 ^^^^. Applying Λ to the super-qudit we obtainThen we perform a projective measurement of the superqudit that projects to the subspaces corresponding to the diagonal blocks given by ^̃^ † ^^^^^̃^^^, i.e., the projective measurement is given by{^^^^}^^^^=−01where ^^^^−1 ^^^^ = ∑  |^^^^^^| ^^=^^^^−1with ^^^^ = ∑^^^^=0 ^^^^.

[0290] Next, we perform a correction operationwhere ^^^^|^^^ = |(^^ −1)mod^^^, which first moves the qudit state into the data subspace and then appliesimplementing. Finally, the outcome of the measurement is "erased" leading to the implementation of the channel, i.e.,Therefore, the detailed procedure implements the quantum channel to the qudit state. However, we still need to show that one always can find a unitary matrix Λ that fulfills Eq. (B3).

[0291] We make use of the properties that Λ is a unitary matrix if and only if it can be written as34296P-EPwhere{|^^^^^}is an orthonormal basis. As|^^^^^corresponds to the ^^ th row of Λ we haveNote that, by construction, it is fulfilled thatfor 0 ≤ ^^, ^^ < ^^.Therefore, as there is no restriction on the rest of the matrix, vectors (columns){|^^^^^} ^^^^=−^1^ can be freely chosen such that {|^^^^^}^^^^=−01form an orthonormal set, meaning Λ is unitary. Appendix C: Bit flip channel

[0292] We briefly analyze the simulation of a bit-flip channel. We aim to implement the bit- flip channel given by ℰ̂(^^) = ^^^^ + (1 − ^^)^^^^^^ (^^1)We consider the simulation by implementing the ^^ gate with probability ^^, see Fig.21(a), andby using an ancillary system, see Fig. 21(b). In Fig. 21(b) the angle ^^ is given by sin2^(^^ / 2) =34296P-EP1 − = exp^{−^^^^ 2 ^^}. We assume that after each gate the system is affected by depo- larizing noise, given byIn this way, we find the Choi fidelity of each implementation is given byrespectively. Note that ^^ is a parameter that we can freely tune. Therefore, for each value of ^^and ^^ we can find the value of ^^ that maximizes ^^∗ = max^^  ^^. In Fig. 21(c) we show that forall values of ^^ and ^^ it is fulfilled that ^^∗^^ ≤ ^^^∗^. Appendix D: Alternative circuits for amplitude damping channel

[0293] The amplitude damping channel is given by Kraus operators,This channel (and in general any channel) can be simulated in a circuit fashion using different implementations, also assuming different computer noises. For instance, using an ancilla qubit system, the damping channel can be implemented as shown in Fig.19(a). In Fig.19(b) we show how the circuit is affected by depolarizing noise after each gate, while in Fig.19(c) we assume the same noise but the controlled-X gate is substituted by a measurement followed by a condi- tioned X gate.

[0294] While the foregoing is directed to embodiments, other and further embodiments may be devised without departing from the scope determined by the claims.

Claims

34296P-EP CLAIMS 1. A quantum simulation method, comprising: receiving, as an input, a specification (902) of a quantum network protocol associated with a quantum network (100), wherein the quantum network is a distributed network comprising a plurality of nodes (10a-f), wherein a local quantum system (12a-f) is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol, wherein the quantum network protocol comprises a sequence of operations, wherein the sequence of operations includes one or more local quantum evolution operations (20a, 20c, 50b), wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes, and wherein (a), (b) or (c), or any combination thereof, is provided, wherein: (a) the sequence of operations of the quantum network protocol includes one or more classical communication operations (40ce), wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network; (b) the sequence of operations of the quantum network protocol includes one or more quantum communication operations (32ab, 52df), wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network; and (c) a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state (14cde, 34de) of K component quantum systems, wherein K is at least two;34296P-EP in response to the receiving, encoding local quantum systems of the quantum network into at least a subset of quantum constituents (70) of a quantum computer (700), wherein each of the local quantum systems is disposed at a respective node of the quantum network during at least a portion of the quantum network protocol, wherein each of the local quantum systems is encoded into at least one quantum constituent; and performing a simulation (950) of the quantum network protocol using the quantum computer, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.

2. The quantum simulation method of claim 1, wherein performing the simulation of the quantum network protocol comprises: determining an effective completely positive map representing an evolution of a first quantum system from a first time to a second time, wherein the first quantum system includes one or more local quantum systems disposed at one or more respective nodes of the quantum network and / or one or more quantum information carriers transmitted between nodes of the quantum network, wherein the evolution of the first quantum system from the first time to the second time is caused at least by at least one of noise acting on the first quantum system and one or more operations of the sequence of operations of the quantum network protocol; and evolving at least one quantum constituent (70) of the quantum computer to simulate the effective completely positive map.

3. The quantum simulation method of claim 2, wherein the effective completely positive map is non-unitary.

4. The quantum simulation method of any of the preceding claims, wherein the quantum network is a noisy quantum network, wherein the quantum network protocol includes a first noisy portion (1010) including target noise, wherein performing the simulation of the quantum network protocol comprises:34296P-EP simulating, using the quantum computer, the first noisy portion of the quantum network protocol.

5. The quantum simulation method of claim 4, wherein simulating the first noisy portion of the quantum network protocol comprises: determining information regarding the target noise, particularly wherein the information regarding the target noise includes a noise parameter of the target noise or a noise parameter of a noisy quantum operation subject to the target noise, wherein the first noisy portion of the quantum network protocol is simulated based on the information regarding the target noise.

6. The quantum simulation method of claim 4 or 5, wherein simulating the first noisy portion of the quantum network protocol comprises: performing a first noisy sequence (1200) of quantum operations (1231-1235) on one or more quantum constituents of the quantum computer, wherein source noise is present during at least a portion of the first noisy sequence of quantum operations, wherein the first noisy sequence of quantum operations simulates the first noisy portion of the quantum network protocol.

7. The quantum simulation method of claim 6, wherein the source noise is different from the target noise, particularly wherein a type and / or a strength of the source noise is different from a type and / or a strength of the target noise.

8. The quantum simulation method of claim 6 or 7, wherein the first noisy sequence of quantum operations is determined by performing an optimization algorithm, particularly an optimization algorithm to minimize a distance between the first noisy sequence of quantum operations and the first noisy portion of the quantum network.34296P-EP 9. The quantum simulation method of any of the preceding claims, wherein the quantum computer is a programmable quantum computer configured for quantum simulation of a plurality of different quantum networks.

10. The quantum simulation method of any of the preceding claims, wherein performing the simulation of the quantum network protocol includes at least one of: (a) measuring (960) at least one quantum constituent to obtain a readout, wherein the readout includes information regarding a property of the quantum network protocol; and (b) preparing a quantum state of at least one quantum constituent (70) of the quantum computer, wherein the quantum state corresponds to another quantum state prepared during the quantum network protocol; and transmitting the at least one quantum constituent prepared in the quantum state to a receiver.

11. The quantum simulation method of any of the preceding claims, wherein: at a time T during the quantum network protocol, one or more local quantum systems are disposed at one or more respective nodes of the quantum network, wherein the one or more local quantum systems are in a quantum state at the time T, at a time T’ during the quantum simulation of the quantum network protocol, the one or more local quantum systems are encoded into at least a portion of the quantum constituents of the quantum computer, wherein at the time T’ at least a portion of the quantum constituents is in a further quantum state that encodes at least one property of the quantum state in which the one or more local quantum systems are at the time T, particularly wherein the one or more local quantum systems include M local quantum systems, wherein M is two or larger, wherein the quantum computer includes M subsystems (80a-f) each including one or more quantum constituents (70), wherein, in the further quantum state, each of the M local quantum systems of the quantum network is encoded into a respective subsystem of the M subsystems.34296P-EP 12. An apparatus (1300) for performing a quantum simulation, comprising: a quantum computer (700), comprising: quantum constituents (70); and a quantum evolution unit (1310) configured to perform an evolution of at least some of the quantum constituents; and a classical computing system (900) configured for: receiving, as an input, a specification (902) of a quantum network protocol associated with a quantum network (100), wherein the quantum network is a distributed network comprising a plurality of nodes (10a-f), wherein a local quantum system is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol, wherein the quantum network protocol comprises a sequence of operations, wherein the sequence of operations includes one or more local quantum evolution operations (20a, 20c, 50b), wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes, and wherein (a), (b) or (c), or any combination thereof, is provided, wherein: (a) the sequence of operations of the quantum network protocol includes one or more classical communication operations (40ce), wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network; (b) the sequence of operations of the quantum network protocol includes one or more quantum communication operations (32ab, 52df), wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network; and (c) a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network34296P-EP includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state (14cde, 34de) of K component quantum systems, wherein K is at least two; and in response to the receiving, encoding local quantum systems of the quantum network into at least a subset of the quantum constituents (70) of the quantum computer, wherein each of the local quantum systems is disposed at a respective node of the quantum network during at least a portion of the quantum network protocol, wherein each of the local quantum systems is encoded into at least one quantum constituent, wherein the quantum computer is configured for performing a simulation (950) of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some of the quantum constituents from a first quantum state to a second quantum state using the quantum evolution unit to simulate the sequence of operations of the quantum network protocol.

13. A method of determining a control layout for a quantum computer (700), comprising: receiving, as an input, a specification (902) of a quantum network protocol associated with a quantum network (100), wherein the quantum network is a distributed network comprising a plurality of nodes (10a-f), wherein a local quantum system (12a-f) is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol, wherein the quantum network protocol comprises a sequence of operations, wherein the sequence of operations includes one or more local quantum evolution operations (20a, 20c, 50b), wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes, and wherein (a), (b) or (c), or any combination thereof, is provided, wherein: (a) the sequence of operations of the quantum network protocol includes one or more classical communication operations (40ce), wherein a classical communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network;34296P-EP (b) the sequence of operations of the quantum network protocol includes one or more quantum communication operations (32ab, 52df), wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network; and (c) a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state (14cde, 34de) of K component quantum systems, wherein K is at least two; and in response to the receiving, determining a control layout for the quantum computer for performing a simulation (950) of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents (70) of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.

14. A control layout for a quantum computer (700) for performing a simulation (950) of a quantum network protocol associated with a quantum network (100), wherein the quantum network is a distributed network comprising a plurality of nodes (10a-f), wherein a local quantum system (12a-f) is disposed at each respective node of the plurality of nodes during at least a portion of the quantum network protocol, wherein the quantum network protocol comprises a sequence of operations, wherein the sequence of operations includes one or more local quantum evolution operations (20a, 20c, 50b), wherein a local quantum evolution operation evolves a local quantum system disposed at one of the nodes, and wherein (a), (b) or (c), or any combination thereof, is provided, wherein: (a) the sequence of operations of the quantum network protocol includes one or more classical communication operations (40ce), wherein a classical34296P-EP communication operation includes transmitting classical information from a first node of the quantum network to a second node of the quantum network; (b) the sequence of operations of the quantum network protocol includes one or more quantum communication operations (32ab, 52df), wherein a quantum communication operation includes transmitting a quantum information carrier from a first node of the quantum network to a second node of the quantum network; and (c) a first local quantum system disposed at a first node of the quantum network includes a first component quantum system, a second local quantum system disposed at a second node of the quantum network includes a second component quantum system, wherein the first component quantum system and the second component quantum system form part of a joint quantum state (14cde, 34de) of K component quantum systems, wherein K is at least two, wherein the control layout includes control instructions for the quantum computer, or information that allows determining said control instructions, wherein the control instructions cause the quantum computer to perform a simulation (950) of the quantum network protocol, wherein performing the simulation of the quantum network protocol includes evolving at least some quantum constituents (70) of the quantum computer from a first quantum state to a second quantum state to simulate the sequence of operations of the quantum network protocol.

15. A data carrier or data carrier signal carrying information representing the control layout of claim 14.

Images